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Calculus

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					2
3
4
5
p     p t
          , ( q, t    0)
q     q t



p     p :t
           , ( q, t   0)
q     q:t



p   q      p q
               , r         0
r   r       r




p r      p r
             , (q 0, s         0)
q s      q s



p r      p s
 :           , ( q, r , s      0)
q s      q r
                                                         7




                         6a 2 b
                                .
                         21ab3


6a 2 b       2a
21ab3        7b 2
                          4 xy 2
                                    .
                         16 x 2 y 3


 4 xy 2         y         1
16 x 2 y 3    4 xy 2     4 xy
                         12 x 2 p 3
                                    .
                         18 xpq


12 x 2 p 3    2 xp 2
                     .
18 xpq         3q
                          (a 3)( x y )3
                                           .
                         (a 3) 2 ( x y ) 2


 (a 3)( x y )3           x y
                             .
(a 3) 2 ( x y ) 2        a 3
                         2 x y 3z
                          y 6z x



2 x y 3z            2 x y 3x        6
                                      1
 y 6z x              y 6z x         6
                                          20 x 5 y 4
                         16 x 2 y 3 :                .
                                           3x 2 y
8



                  20 x 5 y 4                          3x 2 y                         3x 2 y
16 x 2 y 3 :                          16 x 2 y 3                       16 x 2 y 3
                   3x 2 y                            20 x 3 y 4                     20 x 5 y 4
            16 x 2 y 3 3 x 2 y           4x4 3 y 4     4 3      12
                                                                   .
               20 x 5 y 4                5 x5 y 4      5x       5x
                           52a 2 b3 39a 4 b 6         91a 3 b
                                   :                           .
                           69 x 4 y 92 x 3 y 7        115 xy 2


    52a 2 b3 39a 4 b 6        91a 3 b        52a 2 b3 92 x 3 y 7    52 92 91 a 5 b 4 x 3 y 7
            :
    69 x 4 y 92 x 3 y 7       115 xy 2       69 x 4 y 39a 4 b 6     69 39 1155a 4 b 6 x 5 y 3

                           3ab 14 x 2 y ab
                                        :    .
                           4 xy 12a 2 b xy 2


    3ab 14 x 2 y ab              3ab 14 x 2 y xy 2          3 14abx 3 y 3   14 x 2 y 2   7 x2 y 2
                 :
    4 xy 12a 2 b xy 2            4 xy 12a 2 b ab            4 12a 3b 2 xy   16a 2 b      8a 2 b

                              x 3 y 2 p pxy
                                       :       .
                              p 2 xy 2 x 2 p 2


    x 3 y 2 p pxy         x3 y 2 p x 2 p 2         x 2 xp     x3
             :
    p 2 xy 2 x 2 p 2      p 2 xy 2 pxy              p y       y
                          3      5     9
                                         .
                          x      x     x


3      5    9    3 5 9            7
x      x    x      x              x
                          x 3         x 1
                                          .
                           4           4


x 3        x 1         x 3 1 ( x 1)             x 3 x 1            4
                                                                     1.
 4          4              4                       4               4
                                                                                                  9


                           2x 3         2x 5     2x 7
                                                      .
                            4b           4b       4b


2x 3          2x 5         2x 7     2x 3 2x 5 2x 7                    2 x 15
 4b            4b           4b            4b                             4b
                           4a 3b        2a b
                                             .
                             10           5


4a 3b          2a b         4a 3b 2(2a b)              4a 3b 4a 2b             5b        b
  10             5               10                         10                 10        2
                            1     1       a 2 b2
                                                 .
                           6a    3b 2      6ab 2


 1      1        a 2 b2         b 2 1 2a 1 1 ( a 2        b2 )   b2     2a a 2      b2
6a     3b 2       6ab 2                 6ab 2                            6ab 2
        2b 2    2a a 2
               6ab 2
                           x 2y         2x 3y
                                               .
                            x2 y          xy 2


x 2y          2x 3y         y ( x 2 y ) x(2 x 3 y )          yx 2 y 2 2 x 2      3xy
 x2 y           xy 2                  x2 y 2                         x2 y 2
          2 y2     2x2      2 xy
                  x2 y 2
                           2x 1 1              x 2
                                                   .
                             x  2x2             x


2x 1 1               x 2         2 x(2 x 1) 1 2 x( x 2)           4x2     2x 1 2x2           4x
  x  2x2              x                    2x2                              2x2
 6x2    2x 1
       2x2
10




                               p ( x)    an x n       an 1 x n 1 ... a1 x a0 ,
     a0 , a1 ,..., an



ab ac         a (b c).


ax ay bx by                ax bx ay by                  ( a b) x ( a b) y         (a b)( x      y ).

                                  ( a b) 2          (a b)(a b) a 2           2ab b 2 .

                                         ( a b) 2           (a b)(a b) a 2         2ab b 2 .

                                               a2     b2     (a b)(a b).

                              ( a b)3      (a b)(a b)(a b) a 3                    3a 2b 3ab 2      b3 .

                                      ( a b)3         (a b)(a b)(a b) a 3 3a 2b 3ab 2 b3 .

                                 a 3 b3         (a b)(a 2         ab b 2 )
                                        a 3 b3         (a b)(a 2      ab b 2 ).




                                           ( a b) 3          a 3 3a 2 b 3ab 2      b3 .


( a b) 3      ( a b) 2 ( a b) ( a 2            2ab b 2 )(a b)
                a3      2a 2 b ab 2     ba 2        2ab 2    b3
                a3      3a 2 b 3ab 2      b3
                                               ( a b) 3       a 3 3a 2 b 3ab 2 b3 .
                                                                                                       11




( a b) 3    ( a b) 2 ( a b) ( a 2              2ab b 2 )(a b)
                  (a 2     2ab b 2 )(a b)               a3       a 2 b 2ab 2ab 2             ab 2 b3
                  a 3 3a 2 b 3ab 2 b3
     ( a b) 3     ( a b) 2 ( a b) ( a 2                2ab b 2 )(a b)
                                                a2     b2        (a b)(a b).


(a b)(a b)         a a a b b a b b                          a2     ab ba b 2
                           a2        b2
                                                a 3 b3        (a b)(a 2             ab b 2 ).


(a b)(a 2       ab b 2 )        a3        a 2 b ab 2 ba 2          ab 2 b3
                                          a3   b3
                                               a 3 b3        (a b)(a 2            ab b 2 )


(a b)(a 2       ab b 2 )        a3        a 2 b ab 2    ba 2       ab 2      b3
                                          a3   b3
                                                                 (1 2 x)3 .


(1 2 x)3    13 3 13 3 12 2 x 3 1 (2 x) 2                           (2 x) 2
                   1 6 x 3 4 x 2 8 x3
                                                                     13      x3 .


13   x3    (1 x)(12 1 x              x2 )      (1 x)(1 x            x 2 ).


                         x       y
                                   : (x         y)
                         y       x
12

    x       y                          x2        y2           (x       y)        x2        y2         1                ( x y )( x y )           x        y
              : (x            y)                          :                                                                                                  .
    y       x                               xy                     1                  xy          x        y            ( x y ) ( xy )              xy

                                            x2
                                   1           : (x                y ).
                                            y2


            x2                         y2            x2                      (y        x)( y          x)           1        y           x
 1             : (x           y)                          : (x         y)                                                                   .
            y2                              y    2
                                                                                        y2                     x       y        y   2



                                       1         1   1                    1
                                                   : 2                       .
                                       a         b   a                    b2


 1          1   1              1             b a b2 a 2                          b a a 2b2
              : 2                                : 2 2
 a          b   a              b2             ab   ab                             ab b 2 a 2
             b a    a 2b 2                                     ab
              ab (b a )(b a )                                 b a
                                            a                      b          a b
                                                     2         2
                                                                                  .
                                    ab b                   a           ab      ab


        a                 b             a b                  a                    b                   a b
             2        2
ab b              a           ab         ab               b(a b)              a (a b)                  ab
             a2       b 2 (a b)(a b)                               a2        b2 a 2 b2                      2a 2                   2a
                         ab(a b)                                            ab(a b)                       ab(a b)               b( a b)
                                            1               2b                     a b
                                        2
                                                                                          .
                                    a            ab            3
                                                           a ab 2                 2
                                                                                 a b ab 2


        1          2b                         a b                          1                      2b                     a b
    2
a           ab        3
                  a ab 2                    a b ab 2
                                             2
                                                                       a (a b)             a (a   2
                                                                                                       b ) 2
                                                                                                                       ab(a b)
                  1                     2b                                  a b                 b(a b) 2b 2 (a b) 2
             a ( a b)              a (a b)(a b)                           ab(a b)                  ab(a b)(a b)
             ba b 2         2b 2 (a 2 2ab b 2 )                                  ba b 2 2b 2 a 2 2ab b 2
                          ab(a b)(a b)                                                ab(a b)(a b)
                                                                                       13


              ab a 2              a (b a )               1
          ab(a b)(a b)        ab(a b)(a b)           b( a b)
                       ab a   1  3b
                       2
                            :    3
                                    .
                      b b 1 b 1 b 1


 ab a   1  3b                     ab a   1                         3b
2
      :    3                         2
                                       :                           2
b b 1 b 1 b 1                    b b 1 b 1               (b 1)(b        b 1 12 )

     ab a   1                      3b
      2
          :
    b b 1 b 1                (b 1)(b 2 b 1)

      ab a    b 2 b 1 3b
           :
    b 2 b 1 (b 1)(b 2 b 1)
      ab a     b 2 2b 1
           :
    b 2 b 1 (b 1)(b 2 b 1)
     ab a (b 1)(b 2 b 1)
    b2 b 1   b 2 2b 1
    (ab a )(b 1)           a (b 1)(b 1)          a (b 1) 2   a (b 2 2b 1 12 )
                                                                                   a
       b 2 2b 1              b 2 2b 1           b 2 2b 1         b 2 2b 1
                                 1                1
                       1 a                : 1        .
                              1 a               1 a2


            1            1               (1 a )(1 a ) 1   1 a2 1
1 a             : 1                                     :
          1 a          1 a2                  1 a           1 a2
      1 a2 1 1 a2             2 a2 1 a2              1  1 a2
       1 a   2 a2             1 a 2 a2              1 a   1
      12 a 2     (1 a )(1 a )
                              1 a
       1 a          1 a
                      16 x x 2       3 2x       2 3x
                                                     .
                       x2 4           2 x        x 2
14

16 x x 2    3 2x       2 3x             16 x x 2          3 2x         2 3x
 x2 4        2 x        x 2          ( x 2)( x 2)          x 2          x 2
          16 x x 2             3 2x      2 3x           16 x x 2               3 2x     2 3x
       ( x 2)( x 2)            ( x 2)     x 2        ( x 2)( x 2)               x 2      x 2
       16 x x 2    ( x 2)(3 2 x) ( x 2)(2 3 x)
                       ( x 2)( x 2)
       16 x x 2    (3 x 2 x 2 6 4 x) (2 x 3 x 2                   4 6 x)
                           ( x 2)( x 2)
       16 x x 2    (7 x 2 x 2 6) (8 x 3 x 2               4)
                       ( x 2)( x 2)
       16 x x 2    7 x 2 x 2 6 8 x 3x 2              4           x 2                 1
                    ( x 2)( x 2)                             ( x 2)( x 2)           x 2
                        5a         5x           10ax    a               x           2ax
                                                     :                                   .
                       a x        a x          a2 x2   a x             a x         a2 x2


  5a        5x     10ax      a                  x         2ax
                   2    2
                          :
 a x       a x    a x       a x                a x       2
                                                         a x2
            5a        5x              10 x                    a            x            2 ax
                                                     :
           a x    a        x    (a    x )( a    x)       a        x    a       x   (a   x )( a   x)

        5a (a x) 5 x(a x) 10ax   a (a x) x(a x) 2ax
                               :
              (a x)(a x)              (a x)(a x)

        5a 2 5ax 5 xa 5 x 2 10ax   a2                        ax xa x 2 2ax
                                 :
              (a x)(a x)                                     (a x)(a x)

        5a 2 10ax 5 x 2   (a x)(a x)
                        : 2
         (a x)(a x)       a 2ax x 2
       5(a 2 2ax x 2 )          5(a x) 2
                                         .
        a 2 2ax x 2              (a x) 2
                         4a 2               25b 2            2a       5b
                      10ab 25b 2         4a 2 10ab           5b       2a
                                                                                                                        15


   4a 2                25b 2                            2a       5b
10ab 25b 2          4a 2 10ab                           5b       2a
            4a 2                           25b 2                     2a      5b
         5b(2a 5b)                      2a(2a 5b)                    5b      2a
         2a 4a 2 5b 25b 2                          2a (2a 5b)2a 5b(2a 5b)5b
                                                 10ab(2a 5b)
         8a 3 125b3                 4a 2 (2a 5b) 25b 2 (2a 5b)
                                    10ab(2a 5b)
         8a 3 125b3 8a 3 20a 2 b 50ab 2 125b3
                    10ab(2a 5b)
         20a 2 b 50ab 2                      10ab(2a 5b)
                                                         1
         10ab(2a 5b)                         10ab(2a 5b)
                                         1                           2                    1
                                    2                            2                   2
                                                                                               .
                                x        6x 9                x           9       x        6x 9


        1                   2                      1                         1                   2              1
    2                   2                    2                                       2
x       6x 9        x           9        x         6x 9                  ( x 3)             ( x 3)( x 3)     ( x 3) 2
         ( x 3) 2            2( x 3)( x 3) ( x 3) 2
                            ( x 3) 2 ( x 3) 2
         x 2 2 x 3 32 2( x 2 32 ) x 2                                        2 x 3 32
                     ( x 3) 2 ( x 3) 2
         x2    6 x 9 2 x 2 18 x 2                                    6x 9                     4x2
                   ( x 3) 2 ( x 3) 2                                                     ( x 3) 2 ( x 3) 2
                                  3a              2b     3a 2b
                                  3a              2b     3a 2b .
                                9a 2             4b 2    9a 2 4b 2
                                9a 2             4b 2    9a 2 4b 2
16



  3a    2b      3a 2b           (3a 2b) 2 (3a 2b) 2
  3a    2b      3a 2b              (3a 2b)(3a 2b)
9a 2   4b 2     9a 2 4b 2        2
                              (9a 4b 2 ) 2 (9a 2 4b 2 ) 2
9a 2   4b 2     9a 2 4b 2       (9a 2 4b 2 )(9a 2 4b 2 )
                   (3a ) 2  2(3a)(2b) (2b) 2 ((3a) 2 2(3a)(2b) (2b) 2 )
                                       (3a 2b)(3a 2b)
       (9a 2 ) 2    2(9a 2 )(4b 2 ) (4b 2 ) 2 ((9a 2 ) 2 2 9a 2 2 9a 2 4b 2 (4b 2 ) 2
                                     (9a 2 4b 2 )(9a 2 4b 2 )
              (3a ) 2  2(3a )(2b) (2b) 2 (3a ) 2 2(3a )(2b) (2b) 2
                                  (3a 2b)(3a 2b)
       (9a 2 ) 2         2     2
                    2(9a )(4b ) (4b 2 ) 2 (9a 2 ) 2 2 9a 2 4b 2 (4b 2 ) 2
                                 (3a 2b)(9a 2 4b 2 )
             12ab 12ab                            24ab
         (3a 2b)(3a 2b)                           2
                                               9a 4b 2
           72a 2 b 2 72a 2 b 2                   14a 2 b 2
       (9a 2 4b 2 )(9a 2 4b 2 )        (9a 2   4b 2 )(9a 2 4b 2 )
         24ab(9a 2 4b 2 )(9a 2 4b 2 )          9a 2 4b 2
            144a 2 b 2 (9a 2 4b 2 )               6ab
                          x 1        1      2x
                                                .
                        4 x3 x            2
                                   2x 1 4x 4x 1


  x 1           1      2x                x 1             1      2x
4 x3 x               2
              2x 1 4x 4x 1             x(4 x 2 1)     2 x 1 (2 x 1) 2
             x 1                    1      2x
       x(2 x 1)(2 x 1)           2 x 1 (2 x 1) 2
       (2 x 1)( x 1) x(2c 1)(2 x 1) x(2 x 1)2 x
                    x(2 x 1) 2 (2 x 1)
       2x2      2 x x 1 x(4 x 2 1) 2 x 2 (2 x 1)
                     x(2 x 1) 2 (2 x 1)
       2 x x 1 4 x3 x 4 x3              2x2              1
            x(2 x 1) 2 (2 x 1)                  x(2 x 1) 2 (2 x 1)
                                                                                                                     17




 6ac                                              4ab 2 c 3                       7 ab 2 p 3
         ;                                                   ;                               .
14a 3 bc                                         12a 3 b 2 c                       21abp

 (a b)( x 1) 2                                   x(a 2 y )                              x 2 y 2 ( x 1)
                 ;                                         ;                                             ;
3(a b) 2 ( x 1)3                                 2a 2 4ay                         7 x 2 y 3 ( x 2 2 x 1)


    2a b        3a 2b                                     1        1       2b 3a                 x y     x y
                      ;                                                          ;                             .
      4           6                                      3a        2a       6ab                   x2 y    y2 x

   1 9a 2                                                              x4 x                       ( x 3) 2 25
          ;                                                                       ;                              .
    3a 1                                                 ( x2          x 1)( x 1)                (2 x 4)( x 2 1)

      3           4                    2                     x2          x 1     2x 1
                                             ;                                        ;
    x x2        1 x2           x x         2
                                                        x2        36     x 6     x 6
    2x        x(2 x 2 2 x 3)                            x 1
                                                              .
   x 1              x3 1                         x2       x 1
              1                    2                     1
         2                     2                    2
                                                              ;
     x        6x 9         x           9        x        6x 9
     x 1             1                    2x
                                               ;
   4 x3 x          2x 1                    2
                                       4x 4x 1
          1     x4       x                                                     x3 x y 3      y2 y 1 y 1
                           ;                                                                            ;
     x3       1 x2       x                                                     y4 1          x 1    x 1
                            1
                         x2
     2 x                    2                  x                                a b     a c   b2
         1                                       ;                                                .
     x 1                 2x 1                  2                                 b       a a 2 bc


         x2    2           3 1 3x x 2                                            x2 y 2     x4 y 4
                     :                ;                                                 :              .
     x2       x 1        x 1   x3 1                                            1 6m 9m 2 1 18m 2 81m 4
18

          ( a b)3                                                                3a      2b     3a 2b
                      a b
             3ab          ;                                                      3a      2b     3a 2b .
             ( a b) 2                                                          9a 2     4b 2    9a 2 4b 2
                       1
                ab                                                             9a 2     4b 2    9a 2 4b 2
            a           b     a                  2                b
                    2     :                                                    ;
            b           a   ab b 2             a b        a   2
                                                                      ab

            1 1 2a 2                3a 2                          1
                                                      a                    .
          a 1 1 a3                     2
                                   a a 1                      a 1



     3 y 2 ((2 y 1)      y 1) y (1 y                 y2 ) y2           y

     2 x 2 a a (1 2 x 2 ) (a x( x a )).

     (2 x 3 ( x 3 1) ( x 3)2 x 2 2 x 3 )(3 x) 2 3 x 5 .

     ( x 1)(1 x x 2               x3       x 4 ) ( x 1)(1 x x 2                    x3   x4 )
     (x     y z )( x      y ) ( y x z )( z x) ( z x                                y )( y z )
     ( x 1)( x 2)( x 3)( x 4) ( x 2)( x 3)( x 4)( x 5)


     ( y 2)( y 2) ( y 3)( y 3) y (2 y 1) 4

     (5a b 1)(5a b 1) 24(a 1) 2 b 2 1

     ( x 2 y 2 )2 ( x 2 y 2 )2 .

     (2 x 1)(2 x 1)(4 x 2 1).

     (a b c) 2 (a b c) 2 .

     (a 2b c) 2 (a 2b c) 2 .

     (x     y )3 ( x     y )3 .
                    3                  3
      1                  1
        b       y          b       y .
      2                  2

     ( a b c )3 ( a b c )3 .
                                                                                                      19


( a b) 2       2ac 2(a b) 2bc (c 1) 2 .




x2        y2        b             a
                                          .
     ab        x           y x        y

  x2       xy 4 x 3               4 y3                                  a b 2(a b)
                                       .                                       :     .
7 x2       7 y 2 x2               xy                                    a 2 bc   abc

x2        y2                                                             m3 n3 mn           m2
             : (x          y ).                                                    :           .
x2        y2                                                            (mn n 2 ) 2 m 2     n2

14a 4b 2 10 x 2 y 3                                                     9cx 3 2ab 2 4by 2
                    .                                                                     .
 5 xy 21a 2b3                                                           16ab cxy 3ax 2

x2 y 2 4 y 2     x2 y                                                   2 x 3 2 y 3 15 x 2 15 y 2
                        .                                                                         .
    4 xy     2 xy x 2 y                                                  3 x 3 y x 2 xy y 2

a 2 b2     a 4 b4                                                       a 2 ab a 3 b3
                    .                                                                   .
a 2 b 2 a 2 2ab b 2                                                     a 2 b 2 ab(a b)

     ( x 2 y ( x y ))( x 2 y 2 )                                        a 2 2ab b 2 a 3 b3
                                    .                                                      .
( x 3 3 xy ( x y ) y 3 )( x 3 y 3 )                                      a 2 ab b 2 a b

m 2 m(a b) ab m 2 c 2                                                   ab ad ba ad
                      .                                                             .
m 2 (a c)m ac m 2 a 2                                                   bc cd bc cd

   x4 y 4 x y                                                           a 2 b 2 ab x 2 y 2 2 xy
                .                                                                               .
 2   2
x y 2 xy xy x 2                                                            x2 y 2     a 3 b3

   3y2                        y                            1
                                                                    .
x 4 xy 3           x   3      2
                            x y       y x 2
                                                   x   2
                                                               xy

x3        y3                      2y                            2 xy 4 y
                                                                              .
     2y         4 2y               2x         xy           (x     y )( x 2 4)
20

       1                  1                                                                         1               1
     a b              a b a.                                                                        a           b c                 b2    c2 a2
                                                                                                                                1               .
      1                1 b                                                                          1            1                       2bc
     a b              a b                                                                           a           b c

                                                                                                                        2
          2                             1                          2                    1               x           4
                                                                                                                            .
     4x       x2               x2            4x           16           x2          16 4 x               x           4

      ( a b) 2    ( a b) 2
               1           1
         4ab         4ab                                                       (ab (a b) 2 )((a b) 2 ab)
                                                                                                         .
         (a b)3 3ab(a b)                                                          (a b)3 3ab(a b)

      y2          x2 m n                                  x                m n
                                                                               .
      m2          n2 x y                              n       m             2y

     x3 9 y 2 x                         x 3y                       x 3y
                                                                           .
     9 y 2 x2                          x 2 3 xy                   3 xy x 2

              x                          x                    y2        2 xy        x2              y
          2           2                           2
                                                                                                            .
      x           y            (x            y)                         2x                      x       y

     ( a b) 2                           a                         a                 3a b
                                              2               2            2
                                                                                         .
        a                      ( a b)                     b            a             a b

          a                    b              2ab                              a            b                2ab
                                                                                                                 .
      a b                 a b                 2
                                             b a2                      a b              b a                 a b22


                                   2
     1 y2                 xy                   x y                      x          y                x
                                                                                                            .
     x x                  y                  ( x y)2                   xy          y2       x           y
                              ax b




                                                 a z   b.
                                                       b
a   0                                             x      .
                                                       a
        b
    a       b.
        a
a   0            b   0           0 x b, b   0,



a   0, b    0,           0 x 0




            x 4      2 x 7.
22




     x 4          2x 7
     x 2x              4 7
         x        11
     x 11.
              11 4            2 11 7
                       15 22 7
                       15 15.
                              ( x 3) 2( x 1) 5.


     ( x 3) 2( x 1) 5
     x 3 2x 2 5
         x 5 5
         x 0
     x 0.
                              3 x ( x 2) 12( x 3).


     3 x ( x 2) 12( x 3)
     3x      x     2 12 x 36
     2 x 2 12 x 36
         2 36 12 x              2x
     10 x     34
             34        17
     x                    .
             10         5
                                     1          3
                              4y       ( y 1)     y.
                                     2          2


             1                  3
     4y        ( y 1)             y
             2                  2
                                                                          23


           1         1     3
4y           y               y
           2         2     2
       1             3         1
4y       y             y
       2             2         2
8y         y 3y            1
           2               2
4y         1
2          2
      1
y       .
      4
                           x 1            x 1 x
                               x                .
                            3              4  2


                                               x 1       x 1 x
                                                   x
                                                3         4  2
                                                                 2,3, 4
      12.
x 1                  x 1           x
    x                                12
 3                    4            2
4( x 1) 12 x 3( x 1) 6 x
4 x 4 12 x                 3x 3 6 x
    8x 4 9x 3
    17 x         7
       7
x        .
      17
                         3x 1                  1   3
                              2( x 1)            x   ( x 3).
                           5                   2   5


3x 1            1   3
        2( x 1)   x   ( x 3) | 10
  5             2   5
2(3 x 1) 10 2( x 1) 5 x 2 3( x 3)
6 x 2 20( x 1) 5 x 6( x 3)
24

     6 x 2 20 x 20 5 x 6 x 18
     26 x 18          x 18
     27 x        36
                36        4
     x                      .
                27        3

                                     2      1                                                 3
                                x      x      .                                  x 7            x.
                                     3      4                                                 2


              2       1                                                3
         x      x                                          x 7           x
              3       4                                                2

         3x 2 x       1                                          3
                                                           x       x         7
           3          4                                          2
                                                           2 x 3x
         5x      1                                                           7
                   12                                         2
         3       4
                                                             x
         20 x    3                                                     7 | ( 1)
                                                            2
                3
         x                                                 x
                20                                               7| 2
                                                           2
                                                           x 14

                                    1             1                                  x    x      x
                           x          x 6           x 4.                                           .
                                    2             3                                  8    4     12


              1   1                                        x     x x
         x      x   x               4 6                                          0
              2   3                                        8     4 12
         6 x 3x 2 x                                        3x 6 x 2 x
                                    10                                               0
             6                                                24
         x                                                 7x
                10                                                0 | 24
         6                                                 24
         x      60.                                        7x     0 |: 7                 x 0.
                                                                                                25

                                                                                2
                               x 1       x 2      1                     x
                                                                                3    x    1
                                                                                            .
                                5         3       5                         4       12   18

    3( x 1) 5( x 2)                      1             3x 2
           15                           15               3      x      1
                                                         4     12     18
    3 x 3 5 x 10                     1
                                       15                1
         15                         15
                                                       3x 2 x          1
    3 x 3 5 x 10 1                                                       36
                                                        12  12        18
 8 x 13 1                                              3(3 x 2) 3 x     2
 8 x 14 |: 8
                                                       9 x 6 3x     2
      14               7
    x                    .                                                           8   2
       8               4                               12 x   2 6               x          .
                                                                                    12   3

                                    1
                              x 1
                                    3        1   x 2
                                3            4    5


        4
x
        3     1          x 2
    3         4           5
3x 4
  3                1         x 2
  3                4          5
3x 4           1             x 2
                                 180
  9            4              5
20(3 x 4) 45 1 36( x 2)
60 x 80 45 36 x 72
60 x 36 x                    72 80 45
24 x          37
            37
x              .
            24
26

                          2               2
                        x        2x 1
                          3               3   x
                                                1
                          3           1       2
                        1
                          4           5


     3x 2              5
                2x
       3               3       x
                                 1
       1          1            2
       4          5
                      6x 5
     4(3 x 2)           3       x
                                  1
         3              1       2
                        5
     12 x 8     5(6 x 5)        x
                                  1
        3           3           2
     12 x 8     30 x 25         x
                                  1 6
        3           3           2
     2(12 x 8) 2(30 x 25) 3 x 6
     24 x 16 60 x 50 3 x 6
     24 x 60 x 3 x 6 16 50
     81x    72 |: 81
           72   8
     x            .
           81   9
                           2 x 1 3( x 1)( x 1) (3 x 2)( x 1).


     2 x 1 3( x 1)( x 1) (3x 2)( x 1)

     2 x 1 3( x 2 1) 3x 2 3x 2 x 2

     2 x 1 3x 2 3 3x 2           x 2
     2 x 3x 2 3x 2         x    2 3 1
     3x    0
     x 0.
                                                                                27


                          (2 x 1) 2       (3x 1) 2             13x 2 12.


(2 x 1) 2         (3 x 1) 2           13 x 2 12
4x2         4x 1 9x2              6 x 1 13 x 2 12
13 x 2       2 x 13 x 2           12 2
2 x 10
        10
x                5.
         2
                                  x    3x                        x 2
                          1                1
                                  2    2     1 3x                3 3       1.
                              3          2     6                  12


        x     3x                          x        2
1                 1
        2     2     1 3x                  3        3   1
    3           2     6                       12
2 x           3x 2                        x 2
 2              2             1 3x          3          1
 3              2               6          12
2 x 3x 2 1 3x x 2
                         1| 36
  6     4     6     36
6(2 x) 9(3 x 2) 6(1 3 x) ( x 2) 36
12 6 x 27 x 18 6 18 x x 2 36
14 x        46
        46       23
x                   .
        14       7
                                  3 x              x       x
                          1
                                    2              3       2     11
                                           x                        0
                                  5                    5         15


        3 x               x       x
1
          2               3       2     11
                      x                    0
        5                     5         15
28

     2 3 x              2 x 3x
       2                   6       11
                x                           0
       5                   5       15
     x 1   5 x 11
         x                    0
      10   6 5 15
     x 1      x 11
            x         0 | 30
      10      6 15
     3( x 1) 30 x 5 x 22 0
     3 x 3 25 x         22
                              19
     28 x   19. Pra x            .
                              28



                                       A
                                            0     A 0 B     0.
                                       B


                        x 1
                                  0.
                        x 2


     x 1
            0       x 1 0 x 2               0     x 1 0     x    2   x 1.
     x 2
                        x 4
                             0.
                        x2 1


     x 4
          0         x    4 0       x2       1 0    x   4.
     x2 1
                        x( x 3)
                                       0.
                         x2 9


     x( x 3)
                0   x( x 3) 0 x 2 9 0
      x2 9
         ( x 0 ose x 3 0) ( x 3)( x 3) 0
                                                               29

  ( x 0 ose x 3 0) ( x 3 0, x 3 0)
  ( x 0 ose x 3) ( x 3, x 3)  x 0.
              3x 1
                   1.
              2 x


3x 1        3x 1                  3x 1 2 x
     1           1 0                                 0
2 x         2 x                      2 x
  2x 3                                               3
          0        2x 3 0 2 x              0     x     .
   2 x                                               2
              7x 1
                        2.
              2 x


7x 1     7x 1     7x 1 4 2x
      2       2 0             0
2 x      2 x         2 x
  5x 5
        0 5x 5 0 2 x 0   x 1.
   2 x
              3x 1         2x
                              .
              x 3         x 3


3x 1 2 x           3x 1       2x               3x 1 2 x
                                       0                   0
x 3 x 3            x 3       x 3                 x 3
   x 1
       0          x 1 0 x 3 0
  x 3
  x 1 x       3     x 1.


                  x 1   x 3
                                  2.
                  x 3   x 1


x 1   x 3
              2
x 3   x 1
( x 1) 2 ( x 3)( x 3)
                             2
     ( x 3)( x 1)
30

     x2      2x 1 x2 9
                                       2 | ( x 3)( x 1), x       3, x 1.
          ( x 3)( x 1)
     2x2        2 x 8 2( x 3)( x 1)
     2x2        2 x 8 2( x 2        3 x x 3)
     2x2        2x 8 2x2           4x 6
         6x           6 8
                1
     x            .
                3
                              2x 1       3x 1
                                              2 0.
                              2x 3       3x 2


     2 x 1 3x 1
                   2 0
     2 x 3 3x 2
     (2 x 1)(3x 2) (2 x 3)(3x 1) 2(2 x 3)(3x 2)
                                                                               0
                    (2 x 3)(3 x 2)
     6x2        4 x 3 x 2 6 x 2 2 x 9 x 3 2(6 x 2                        4 x 9 x 6)
                                                                                        0
                             (2 x 3)(3 x 2)
     12 x 2 18 x 5 12 x 2 26 x 12                                                     3       2
                                                     0 | (2 x 3)(3 x 2), x              , x
             (2 x 3)(3 x 2)                                                           2       3
     8x 7             0.
              7
     x          .
              8
                                 1 x            5 x          1
                                                                     .
                              2 x( x 2)        2x2 4x    x 2


        1 x                 5 x           1
     2 x( x 2)             2x2 4x        x 2

        1 x                   5 x          1
                                                 0
     2 x( x 2)             2 x( x 2)     x 2
     (1 x)( x 2) (5 x)( x 2) 2 x( x 2)
                                                                 0
               2 x( x 2)( x 2)
                                                                                  31


  ( x 2)(1 x (5 x)) 2 x 2         4x
                                           0
          2 x( x 2)( x 2)
  ( x 2)(1 x 5 x) 2 x 2         4x
                                       0
         2 x( x 2)( x 2)
   4x 8 2x2 4x
                         0
   2 x( x 2)( x 2)
        2x2 8
                       0 | 2 x( x 2)( x 2), x       0, x       2, x   2
  2 x( x 2)( x 2)
   2 x 2 8 0 |: ( 2)                           x2    4 0.
      x2   4 0




                                                           x 24.

x ( x 24) 100       2 x 124                    x    62.
                                                                      62 24 48.




                                                     x     x
32

                                                      1
                                                 x   x   x
                                                      2
                                                 1      1
                                            x x     x      x
                                                  2     4
                                            1   1
                                        x x   x   x 1
                                            2   4


               1   1
     x x         x   x 1 100
               2   4
           x    x
     2x             99
           2    4
     8x 2x x
                    99
        4
     11x
           99
      4
                                x 36.
                         36




           x x       x   x
      x                5   4
           6 12      7   2
                              x 84.
                                              33


                                 3
                                   .
                                 5

                                        9
                                          ?
                                       10

                           9
                             ?
                          10




3 x 9
5 x 10
10(3 x) 9(5 x)
30 10 x          45 9 x
    x 15.


3            9
     x
5           10
      9      3
x
     10      5
      3
x       .
     10
34




     2 3 x 5.
     3x 4              2 x 6.
     x 3 3 x 1.
     2 x (2 x 3)                4 x 1 2 x 3 (7 2 x).
                                              x
     0.2 ( x 1) 0.5(3x 9)                              2.
                                              3
     x       2x 1   x
                  1   .
     3         6    3
                      1 3x            2x 1
     7 2x                        2         .
                        7               3
     3 x 1 13 x                  7 x 11( x 3)
                                              .
        5    2                    3      6
     7 9x                   2 x
                       1               7 x.
       4                     9
         x 5          x 7        x 6
                                         3.
          4            2          3
         3( x 1)           2( x 3)                     31x
                                       2x 1                   4.
             2                 5                       10
                     3y 1       2x 7              x 6
         x                                5           .
                       5          2                2
                 x    7x                      7
      1                   1                       6x
                 4    2     1 5x              2              1
                                                               .
             2          6    24                 12           3
                 6 x             x    3 x
      1
                   3        x    2      4         3.
                 2                    2
         x 3          3x 1
                                 2.
         x 3          3x 1
                                                                                          35


2x 9                  3x
                                   2.
2x 5                 3x 2
4x 1   5x                              6x 4                  x 1
                                                      1          .
x 4 3 x 12                             5 x 20               2x 8
 x 1   11x 5                                x 3           1
                                                            .
2 x 1 12(2 x 1)                            4 8x           6
        1              5 x                 7            x 1
                                                               .
8 x 16                8x 4x2               8x        2 x( x 2)

6x 5                 7 3x          12 x 2 30 x 21
                                                  .
4x 3                 3 4x              16 x 2 9
  1                          1         3(2 x 1)
                                                          0.
4x 6                 8 x 12             4x2 9
                1                      1              3( x 1)        1
            2                      2
                                                                          0.
18 x                30 x 12 x              20 x      18 x 2 50       6x
        2                        1                   1
    2                    2                       2
                                                                0.
x               4    x           4x 4        x       5x 6




                                                                               ( x, y )
                    3 3 3 3.


                                            3 3 3 3 81.
                                                          3 3 3 3
 4
3.

                     34        3 3 3 3.
     4
     3
                         34.
                                   34.
          34



3 3 3 3
34




          n    N.                            a
                         n
                     a         a a ... a .
                                 n faktor
                                                                                            37



                                                      1 1 1
        2 2 2 2 2;                                          ;
                                                      2 2 2
        3
          ;                                           3 3 ... 3;
        4                                                 n faktor


        1 1    1                                          1              1            1
            ... ;                                                               ...     .
        3 3    3                                          4              4            4
          m faktor                                                       t faktor




                           2

                         2 2 2 2 2               25.
1
2
                                                 3
                          1 1 1             1
                          2 2 2             2
3
4
                                            1
                               3           3
                                             .
                               4           4


                          3 3 ... 3 3n.
                               n faktor


1
3
                                                  m
                                             1
                         3 3 ... 3                    .
                          m faktor           3
    1
    4
                                                                     t
                     1     1                 1                1
                                     ...                        .
                     4     4                 4                4
                          t faktor
38


                                                                           1
                                                                       1
                 36 ;                                 ( 2)5 ;            ;
                                                                       3
                 4n ;                                 5m ;         ( 6) s .


                          3                            6               3       6

                                       36     3 3 3 3 3 3.
         2                         5
                               5
                        ( 2)           ( 2) ( 2) ( 2) ( 2) ( 2).
             4
             3
                                                  1
                                              4              4
                                              3              3
     4                        n
                                         4n    4 4 ... 4.
                                                  n faktor


     5                        m
                                         5m    5 5 ... 5.
                                                  m faktor


         6                         s
                              ( 6) s        ( 6) ( 6) ... ( 6)
                                                      s faktor
                                                                                                                       39




1 1 1 1                                           1 1 1
        .                                     1    1 1 .                              0.7 0.7 0.7 0.7 0.7.
4 4 4 4                                           3 3 3
                                                   2       2            2         2       2       2              2
( 0.4) ( 0.4) ( 0.4).                                                                                              .
                                                   3       3            3         3       3       3              3
1 1    1                                           1       1                1
    ... .                                                    ...              .
2 2    2                                           3       3                3
     n faktor                                          m faktor




                                      4                                                                  3
                                  1                                                                  1
( 2)5 .                                   .                       ( 0.5)2 .                      1           .
                                  3                                                                  2
            1                             n
       7                              3
         .                                    .                   6n 1.                       ( 3) n s .
       3                              2


                                                            1
                                                       3          3
                                                                    .
                                                       4          4


                                                            1
                                                       4             4
                                                                       .
                                                       3             3



                                                       a1       a.


                           17 ;                                 ( 1)6 ;                         ( 1)7 .




17     1 1 1 1 1 1 1 1.
                7 faktor
40

     ( 1)6       ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 1.
                         1              1              1

            7
     ( 1)        ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1)                        ( 1) 6 ( 1) 1 ( 1)   1.
                                   6 faktor




     1n     1,
     ( 1) n      1,
     ( 1) n         1,


                               ( 3) 4                              ( 3)3 .


                      ( 3) 4


          ( 3) 4         ( 3) ( 3) ( 3) ( 3)
                       ( 1) 3 ( 1) 3 ( 1) 3 ( 1) 3
                      ( 1) ( 1) ( 1) ( 1) 3 3 3 3 ( 1) 4 34
                                              ( 1) 4   1.
                                                       ( 3) 4      34.
                                                             ( 3)3 .


            ( 3)3        ( 3) ( 3) ( 3) ( 1) 3 ( 1) 3 ( 1) 3
                         ( 1) ( 1) ( 1) 3 3 3 ( 1)3 33
                 ( 1)3         1
                                               ( 3)3        1 33         33.


                                                                    ( a)n      an .
                                                                    ( a)n       an .
                                                                                                                                   41




                             115 ;                           112 ;                            ( 1)15 ;                  ( 1)14 ;
                               ( 1)13 ;                       ( 1)12 ;                        112    ( 1)11 ( 1)10 ;
                             14        ( 1)3 ( 1)1 ( 1)7 .


           1                                                      1                       115       1.
112   1.
           15                                                                         1
                         1               ( 1)15              1.
( 1)14         1.
 ( 1)13             ( 1) 1.
 ( 1)12             1.
112   ( 1)11 ( 1)10                    1 ( 1) 1 1 1 1 3.
14    ( 1)3 ( 1)1 ( 1)7                            1 ( 1) ( 1) ( 1) 1 1 1 1 2.


                             ( 2)3 ;                                       ( 2) 4 ;                            ( 2)5 ;
                                         3                                        2                                 4
                                     1                                        3                                 1
                                             ;                                        ;                                 ;
                                     4                                        2                                 3
                             ( 2)3 ( 2) 4                    ( 3) 4        ( 3)3 ;
                                         3               2             1
                                 1                   1            1
                               1                   1                ;
                                 2                   2            2
                                         2                             2                  3
                                  1                  1            1               1
                                                                                              .
                                  4                  4            2               2


           3                                                         ( 2)3            23            8.
           4                                                         ( 2) 4       24              2 2 2 2 16.
 ( 2)5              ( 25 )        25         32.
42
                   3                 3
           1                 1                1 1 1                      1
                                                                            .
           4                 4                4 4 4                      64
               2               2
        3               3                3 3         9
                                                       .
        2               2                2 2         4
               4             4
       1               1                 1 1 1 1                  1                        1
                                                                                              .
       3               3                 3 3 3 3               3 3 3 3                     81
     ( 2)3 ( 2) 4                    ( 3) 4         ( 3)3               23        24          34       ( 33 )     8 16 81 27 30.
               3                 2              1              3                  2
        1                1                1            3                  3               1           3 3 3       3 3          1
      1                1
        2                2                2            2                  2               2           2 2 2       2 2          2
               27       9            1     27 9 2 1 4                           27 18 4                     5
                                                                                                              .
               8        4            2         8                                   8                        8
               2                              3                    3                  2                                            3
        1                  1              1                1                  1               1       1 1 1                1
        4                  4              2                2                  4               4       2 2 2                2
               1 1 1 1 1 1 1                                              2            1          1    1      1    2
               4 4 4 8 8 16 4                                             8           16          4    4     16    4
                1 1 1 18 9
                            .
               16 2  16  16




      113.                                                         110.                                                        ( 1)7 .
      ( 1)6 .                                                           ( 1)5 .                                                 ( 1) 4 .
      14       ( 1)3         ( 1) 2 .                              18 ( 1)9                     ( 1) ( 1)3 .



      ( 3) 2 .                                                         ( 2)5 .                                                  ( 3)3 .
                                                                                  3                                                        3
                                                                          1                                                            1
      ( 0.5)2 .                                                                       .                                                        .
                                                                          4                                                            2
                                                                                            3               2          2
               2               3                3     2                             1                   3         1
      ( 2)             ( 2)              ( 3)        3.                           1                                        .
                                                                                    2                   2         4
                                                                                                                                                 43


                                                             23 24.


23 2 4       (2 2 2) (2 2 2 2)                                             2 2 2 2 2 2 2.
                  3 faktor                           4 faktor                       7 faktor

                 23 2 4                 23       4
                                                         2 7.


                                                                 a
                                                                am an          a m n , m, n              N.


                                         33 35 ;                                    4 2 43 ;                                     52 54 ;
                                           x 2 x9 ;                                 a a9 ;                                       ( a b) ( a b) 4 ;
                                                     3           2                                   0                   5
                                           x              x                             x        y           x       y
                                                                     ;                                                       .
                                           2              2                                  3                   3




  33 35          33    5
                                   38.
  4 2 43          42       3
                                    45.
  52 54           52       4
                                   56.
   x 2 x9             x2       9
                                       x11 .
                 a1        a,                                a a9          a1 a 9       a1   9
                                                                                                     a10 .
  ( a b) ( a b) 4                          (a b)1 (a b) 4                      ( a b)1           4
                                                                                                      ( a b) 5 .
         3             2                     3 2                 5
    x             x                    x                     x
                                                                     .
    2             2                    2                     2
                  0                          5                           0 5                 5
    x        y                 x       y                 x       y             x        y
                                                                                                 .
         3                         3                         3                      3
44




         32 31 .                                             2 25.                                         20 21.
         A3 A5 .                                             (a 1) 2 (a 1)3 .                              (a b) 2 (a b).
               5             2                                              0                5
           x         x                                        x        y         x       y
                                 .                                                               .         ( x 1)0 ( x 1)0 .
           3         3                                             2                 2



                     x 2 x5                                        x7 .             x7

                                          x7          x x2 x4 , x7          x3 x 4 , x7          x x6 .


                                                              29
                                     29      23      3 3
                                                           23 23 23             8 8 8 64 8 512.
                                             2   9
                                                                                                     2 2 27.


                                      28 ;                                  210 ;                              35 ;
                                      37 ;                                  24 32 ;                            43 32.


     28        22   2 2 2
                                     22 22 22 22             4 4 4 4 16 16                       256.
                                                              16       16

     210       28 22             256 4 1024.
      5        2 2 1
     3         3                 2
                             3 32 3 9 9 3 81 3 243.
     37        35 32             243 9           2187.
     24 32          22 22 32                 4 4 9 16 9 144.
     43 32          42   1
                             32           42 4 9 16 36 576.
                                                                                                                        2.
                                          64 23 ;                           16 128;                            100 28.
                                                                                                          45


64 23    16 4 23              2 4 2 2 23        24       2 3
                                                                    29.
16 128 16 2 64 24 2 26                              24    1 6
                                                                     211.

                                                                                                     100, 28
                                                                               2.

                                                                                     2.
                                                                2      4      7
100 28 128 2 64                       2 4 16          2 2 2                  2.
                                                                                                 3.
                          27 32 ;                                81 35 ;              60 21.




27 32    33 32         35.
81 35    9 9 35         32 32 35           32       2 5
                                                               39.
60 21 81 9 9 32 32                       32     2
                                                          34.


                               2           3.
                          36 72;                                108 36;                   64 6 12.


                                                                    36
                                                36             22 32.
                                                                72.
                                                72             23 32.


36 72     22 32 23 32                 22 23 32 32                   25 34.
        108 36 3 22 32 3 22 33 ; 36                                        22 32 ,
                                 36

             2     3      2     2
108 36       2 3 2 3                  24 35.
        64       26 ; 6        2 3; 12 22 3
64 6 12      26 2 3 22 3 26 2 22 3 3 29 32.
46




     27.                             29.                             43.                       36.
     38.                             23 3.                           34 24 ;                   20 23 32.

                                                                               2.
     16 25.                                  8 23.                                  128 128.
     64 32.                                  64 32.                                 30 34.
                                                                               3.
     81 3.                                   9 34.                                  243 33.
     27 81.                                  27 81.                                 50 31.
                                                                                                2      3.
     24 9.                                   48 27.                                 48 54.
     24 27 36.                               144 12.                                18 36.

                                                          45 : 43.


                      45       4 4 4 4 4
           45 : 43                                 4 2.
                      43         4 4 4


             45
45 : 43              45   3
                              4 2.
             43


                          a                                                            m, n    N
                                               am
                                                          am n .
                                               an
                                                                           m    n

                                                          am
                                              am    m
                                                             .
                                                          am
                                                                                                                                47


                                                               am       m
                                                                            a0


                                                                   am
                                                                            1
                                                                   am


                                                      a
                                                                       a0    1.


                                    27
                                       ;                               36 : 3;               a5 : a 2 ;
                                    24
                                                                                                         3          0
                                        11                                       4    3         a2             a2
                                  a : a;                               ( a b) : ( a b) ;                     :          .
                                                                                                3              3


27
       27   4
                     23.
24
 36 : 3 36 : 31              36     1
                                             35.
a5 : a 2        a5   2
                             a3 .
a11 : a     a11 : a1           a11 1             a10 .
( a b) 4 : ( a b)3                  ( a b) 4          3
                                                          (a b)1            a b.
       3                 0                   3                 3
  a2         a2                   a2                      a2
           :                                     :1                .
  3          3                    3                       3




  26                                                       35
     .                                                        .                            a12 : a 5 .
  24                                                       32
                                                                                                         5                  1
  (ax)9                                                    (b x) 4                           a b               a b
        .                                                          .                                         :     .
  (ax)8                                                    (b x) 4                            2                 2
48
                 4               0                                                  5
         a2            a2                                             x        y                 x       y
                     :               .                                                  :                    .
         4             4                                                  3                          3


                                          4 7 : 43 ;                               65 : 6 4 ;                                162 : 43.


      4 7 : 43        47   3
                                     44       42 42             16 16              256.
     65 : 6 4        65    4
                                 61         6.
                           2
                      16             16 16             42 42              42
     162 : 43                                                                       42       1
                                                                                                     41          4.
                       43              43              42 4               4




       47
          .                                                         65 : 6 6.                                                   79 : 78.
       42
       16                                                           642                                                         81
          .                                                             .                                                          .
       42                                                            46                                                         33



                                         (43 ) 2 .                                                   (43 ) 4 .



                                           3 3
     (43 ) 2         43 43           42 mbledhes         43 2        4 6.
                                                       3 3 3 3
     (43 ) 4         43 43 43 43                     4 4 mbledhes      43 4         412.
                           4 faktor


                                                          a
                                                                                            n mbledhes


                               (a m )n         a m a m ... a m                     am       m m ... m
                                                                                                                  a m n , m, n       N.
                                                         n faktor




                                a) (32 )5 ;                           b) (a 2 ) 4 ;                                   c) (a 5 )0 ;         d) ( x5 )1 ;
                                                                                                                                        49

                                                                                                             3    3
                                         1 16                                    3 2                   x
                             e) ( A ) ;                               f) (( x y ) ) ;           g)                    .
                                                                                                       3


(32 )5          32 5     310.
(a 2 )4         a2 4      a8 .
(a 5 )0         a5 0      a0        1.
 ( x5 )1        x5 1      x5 .
( A1 )16         A116        A16 .
(( x       y )3 ) 2     (x        y )3 2        (x           y )6 .
            3   3              33               9
       x                 x                 x
                                                    .
       3                 3                 3


                                     ( a 5 a 4 )3 ;                                    a ( a 2 )5 ;
                                     (a 3 )2                                           a (a 2 a 3 a 4 )4
                                              ;                                                          .
                                     ( a 2 )3                                                a12


( a 5 a 4 )3           ( a 5 4 )3        ( a 9 )3            a9 3       a 27 .
a ( a 2 )5            a a2 5         a a10              a1 10          a11 .
(a 3 )2         a3 2         a6
                                      a6    6
                                                        a0       1.
( a 2 )3        a2 3         a6
 a (a 2 a 3 a 4 )4                  a (a 2 3 4 )4                     a(a9 )4    a a 36       a 37
                                                                                                      a 37   12
                                                                                                                          a 25 .
       a12                             a12                              a12       a12         a12




  (34 ) 2 .                                     (25 )0 .                                (26 )1 .                            (50 ) 4 .
50

                                                      3     4
                    2 2                        x
       ((a b) ) .                                               .                      (( x 1)1 ) 2 .               ((a 1)0 )1 .
                                               2

                                                                       a

       (a1 a 5 ) 4 .                                      ( a 2 a 7 )5 .                                    (a 0 a1 )0 .

       ( a 5 )3                                           a (a 0 a1 ) 4 a 5                                 a0 a9
                .                                                           .                                     .
       ( a 3 )5                                               a6 a7                                         a9 a0

                                                            (2 3)3 .


          (2 3)3          (2 3)(2 3)(2 3)                  (2 2 2)(3 3 3)                     23 33.


           ( a b) n        (a b) (a b) ... (a b)                           (a a ... a ) (b b ... b)               an bn .
                                     n faktor të ab                        n faktor të a së    n faktor të b së




                                                           a, b
                                                 ( a b) n              a n bn , n       N.



                                    (23 34 )5 ;                                 (2ab 2 c3 ) 4 ;                   ( 2 x 3 y 5 z 7 )3 .


     (23 34 )5         (23 )5 (34 )5        23 5 34 5                215 320.
     (2ab 2 c3 ) 4        24 a 4 (b 2 ) 4 (c3 ) 4                   24 a 4 b 2 4 c3 4          24 a 4 b8 c12 .
     ( 2 x 3 y 5 z 7 )3     ( 2)3 ( x3 )3 ( y 5 )3 ( z 7 )3                    ( 2)3 x3 3 y 5 3 z 7 3
                              ( 2)3 x9 y15 z 21.




       (24 32 )3 .                                        (2 xy 2 z 3 )5 .                             ( 3x 2 5 y 3 )4 .

       (( x    y)2 ( x      y )3 ) 4 .                    (34 43 52 71 90 )5 .                           (2 32 53 7 4 )5 .
                                                                                                                                   51

                                                        4
                                                   3
                                                   2



               4
       3              3 3 3 3                      3 3 3 3                    34
                                                                                 .
       2              2 2 2 2                      2 2 2 2                    24


                                                                                 n faktor të a së
                                           n
                                   a                a a     a                    a a ... a              an
                                                        ...
                                   b                b b     b                    b b ... b              bn
                                                                          a      n faktor të b së
                                                       n faktor të
                                                                          b




                                                              a, b                                  b   0,
                                                                      n
                                                             a                an
                                                                                 , n        N.
                                                             b                bn


                                           5                                                5                              4
                                  23                                                 4 x2                     ( x 2 y)2
                                               ;                                                ;                              .
                                  34                                                 3 y3                     ( x 2 y )3


       5
23                 (23 )5       23 5       215
                                               .
34                 (34 )5       34 5       320
           5
4 x2                (4 x 2 )5      45 ( x 2 ) 5               45 x 2 5                45 x10
                                                                                             .
3 y3                (3 y 3 )5      35 ( y 3 )5                35 y 3 5                35 y15
                      4
( x 2 y)2                   (( x 2 y ) 2 ) 4                 ( x 2 y)2 4                ( x 2 y )8
                                                                                                    .
( x 2 y )3                  (( x 2 y )3 ) 4                  ( x 2 y )3 4               ( x 2 y )12


                                               3              3                                               3
                                       1                 1                                       1       36
                                  2                : 3            ;                             1 :               .
                                       2                 3                                       5       10
52


              3                     3                        3              3                            3                   3                       3
         1                  1                        5               10                 5 10                        5 3                      3            33        27
     2            : 3                                            :                       :                                                                             ;
         2                  3                        2                3                 2 3                         2 10                     4            43        64
                                                                                                                        3
                                    3                                           3            1                  2
      1            36                                6               36                     6                10
     1 :                                               :
      5            10                                5               10                     51               36 6
                                                                     3                  3
                                                                 2                  1            ( 1)3                1
                                                                                                                         .
                                                                 6                  3             33                  27




                                                                                                 4                                                              3
              4 7           2 5                                                         25                                                           4 x2 y 4
         (2 ) : (3 ) .                                                                               .                                                              .
                                                                                        34                                                            5z5
                                    5                                                            0                                                              2
             ( x a)2                                                                    x7                                                           7 4 85
                                         .                                                           .                                                              .
             ( x a )3                                                                   y8                                                              36


                    4                        5                                                                         2                 3       4
               1            3                                                                                     1            1
             1          : 1                      .                                                              1            2                       .
               3            4                                                                                     5            4

                        2       3                        5                                                                       4   5
                3                       1                                                                       3    1
              1                     : 2                      .                                                 2 : 2                     .
                4                       3                                                                            2

                                                                                                                      4 3 : 4 5.




                                    43
         43 : 45                                     43      5
                                                                     4 2.
                                    45



                                    4 2?
                                                                                                                    4 3 : 4 5.
                                                                                                                 53


                                                       43        4 4 4                             1      1
                                      43 : 45                                                                .
                                                       45      4 4 4 4 4                          4 4     42


                                                                  43 : 45       4    2




                                                                                1
                                                               43 : 45             .
                                                                                42
                                  2        1
                              4               .
                                           42


                                                       a
                                                                    n        1
                                                              a                 ,n           N.
                                                                             an
                                                                                             3
                                                                                         2           ( 2) 3 ?


               3           1               1                       3           1                 1
           2                                                ( 2)
                           23              8                                 ( 2)3               8



                                                                    n
                                                               a             ( a) n ?


                                                                                              xn 9
                                       (21      3n
                                                     ) 3;                                           .
                                                                                              x2n 3


(21   3n
           )   3
                        2(1   3 n ) ( 3)
                                                2    3 9n
                                                             29 n 3.
xn 9
                   xn   9 (2 n 3)
                                           xn   9 2n 3
                                                              x    n 6
                                                                         .
x2n 3
54

       45                                             x2                                      ( a b)3
          .                                              .                                            .
       49                                             x7                                      ( a b)5
                      3                                                                                       1
             4 xy                                     (3 2 ) (2 3 )                            4x 1 y 1
            3 2 4
                          .                                         .                                             .
           4 x y                                      (2 4 ) (3 3 )                             2 2 xy
                                                                                                                      2
           1 n   2                       2n 3                     4n 2                         2n 4 3n    3
                                                                                                              5n 2
       (3 ) .                                 .                        .                                                  .
                                         2n 4                     43 n                         2n 1 3n    1
                                                                                                              5n 3



                                  103 36 65 (5 4 ) 4
                                                          .
                              (( 12) 4 ) 4 5 12 3 1 ( 7)0


          103 36 65 (5 4 ) 4                         (2 5)3 36 (2 3)5 5 4 4
     (( 12) 4 ) 4 5 12 3 1 ( 7) 0                      ( 12) 4 ( 4) 5 12 3 1
       23 53 36 25 35 5 16                        23 25 36 35 53 5 16                   23 5 36 5 53 16
         ( 12)16 5 12 3 1                          ( 1)16 1216 5 12 3 1                (3 4)16 5 12 3 1
          28 311 5 13                    28 311 5 13                  28 311 5 13             28 311 5 13
       316 416 5 12 3 1             (22 )16 316 3 1 5        12
                                                                     232 316 1 5 12           232 315 5 12
                                                                    1 1 1               1
      28   32
                 311 15 5 13     ( 12)
                                            2   24
                                                     34 51                                   .
                                                                   224 34 5            24
                                                                                      2 34 5
                               23 ( 7) 4 (3 2 ) 3
                                                   .
                              (4 2 )3 492 (9 2 ) 1


      23 ( 7) 4 (3 2 ) 3                 23 ( 1) 4 7 4 3 2 ( 3)            23 7 4 36
     (4 2 )3 492 (9 2 ) 1                 4 2 3 (7 2 ) 2 9 2 ( 1)          4 6 7 4 92

           23 36                23 36            23 36
                                                             23 (   12)
                                                                           36   4
                                                                                    215 32.
       (22 ) 6 (32 ) 2         2 12 34          2 12 34
                               81 x 4 y 3 ( z 2 ) 1
                                                         .
                              27 x 7 ( y 3 ) 2 ( z 1 ) 2
                                                                                                                                     55


 81 x 4 y 3 ( z 2 ) 1                                     34 x 4 y 3 z 2( 1)                          3 x 4 y 3 z2
27 x 7 ( y 3 ) 2 ( z 1 ) 2                                33 x 7 y 3 ( 2) z 1 2                        x7 y6 z2
           4 ( 7)           3 ( 6)
    3 x                 y                    z2      ( 2)
                                                                    3 x3 y 3 z 4 .
                                        2                   5       3
                                    a                 a
                                        4
                                                 :          7
                                                                         .
                                    b                 b



                            3
                                                                                                                       a2
     2              5                        2                  5        3            2     5 ( 3)       2        15
 a             a                         a                (a )                   a a                   a a             b4
          :                                           :                            :                      :
 b   4
               b    7
                                         b   4
                                                          (b 7 )         3
                                                                                 b4 b      7 ( 3)
                                                                                                       b 4 b 21        a15
                                                                                                                       b 21
    a 2 b 21                2 15
                        a               b 21 (       4)
                                                                a   17
                                                                             b 25 .
    a15 b 4
                                                                                                       x3 y 2 ( x 1 y 3 )4 x ( y 3 )2
                                                                                                  A
                                                                                                          ( x 2 y 3 )2 ( x y 3 )2
                                                                                                                       1
                                                                                                           x 4, y         .
                                                                                                                       2


         x3 y 2 ( x 1 y 3 )4 x ( y 3 )2                                               x3 y 2 ( x 1 )4 ( y 3 )4 x ( y 3 )2
A
            ( x 2 y 3 )2 ( x y 3 )2                                                      ( x 2 )2 ( y 3 )2 x 2 ( y 3 )2
         x3 y 2 x           4
                                    y 12 x y                6
                                                                         x 3 4 1 y 2 12       6
                                                                                                      x0 y   20


            x4 y            6
                                    x2 y 6                                  x4 2 y 6 6                x6 y   12


          6        20 ( 12)                  6
      x        y                        x            y 8.
                            1
     x        4, y
                            2
                                8
                        1               1                                    1 8           28          28              1       1
A (4 6 )                                   (2 1 )               8
                                                                                2                             2    4
                                                                                                                                 .
                        2               46                                   46           (22 )6       212             24     16



                                              x x2                                 1 x1    1 x1
                                                                                                 .
                                            x2 x1 1                             x 2 2x 1 1 1 x 1
56

                                                   1                 1      1
                                              x                1          1
     x   x2           1 x1    1 x      1
                                                  x2                 x      x
     2
 x       x1 1     x 2
                       2x 1
                            1 1 x      1
                                             1    1
                                                     1
                                                          1
                                                                   2
                                                                      1
                                                                        1 1
                                                                            1
                                             x2   x       x2          x     x
          x3 1           x 1       x 1
           x2             x         x          x3 1       x( x 1)      x 1
         1 x x2        1 2x x2     x 1       x2 x 1      2
                                                         x 2x 1        x 1
           x2             x2        x
         ( x 1)( x 2 x 1)   x( x 1)    x 1
             x2 x 1         ( x 1) 2   x 1

                   x     x 1     ( x 1) 2 x x 1     x2   2x 1 x        x2    3x 1
         ( x 1)                                                                   .
                  x 1    x 1         x 1    x 1          x 1                x 1
                                                                                                    57


2. RRËNJËZIMI – FUQIA ME EKSPONENT NUMËR
RACIONAL




                                         xn              a




                                     x           ( n a )n                  a
                                 n
                                     a


                                             a2 | a |
     a   0           m, n
                                             m
                                                          n
                                         an                   am .

             a, b                                                                            m, n


                                         n                    n            n
                                             ab                       a        b
                                                              n
                                         n
                                             a                    a
                                             b                n
                                                                  b


                                                 3                5             3       13
                    36   4 49                        8                32            1         1;
                    16       1
                         3                   3
                                                     64                   25        ( 3) 2 ;
                     9       27

                    25       3                                        3             5
                                     8               ( 9                   27           32).
                    36
58
                                                                                                                        3            5
      36 4 49                        3
                                         8        5
                                                       32        3
                                                                         1       13
                                                                                          1 6 4 7                           23            25 1 ( 1)

                                                                                                6 28 2 2 2 36.
                                                                                                                    3
      16            1                                                                             16                1
                3                    3
                                         64                25                ( 3) 2                                              3
                                                                                                                                     43     5 | 3|
       9            27                                                                             9            3
                                                                                                                    27
                                                                                                4        1
                                                                                                           4 5 3 3.
                                                                                                3        3

       25                                                                                                25
                    3
                         8               ( 9           3
                                                               27        5
                                                                                 32)                                3
                                                                                                                        ( 2)3             (3   3
                                                                                                                                                   ( 3)3      5
                                                                                                                                                                  ( 2)5
       36                                                                                                36

                                                                                                  5                                                   101
                                                                                                          ( 2)                  (3( 3)( 2))               .
                                                                                                  6                                                    6


                                              8            32;                                   3 6 2 8;                                           xm    2
                                                                                                                                                                  xm 2 .


      8        32             4 2                 16 2                   4            2         16        2             2       2 4        2

                             8( 2) 2                  8 2 16.

     3 6 2 8                 3 2 3 2 4 2                                     3 2            3 2 4                       2    6( 2) 2           3 2

                             12 2                 3            24 3.

      xm   2
                        xm   2
                                             xm    2
                                                           xm    2
                                                                                 xm       2 m 2
                                                                                                              x2m               ( x m )2 | x m | .
                                         1            1              2           1          2

                                     33           23            33           63           23 .



       1        1                2           1             2
                                                                                                                            3
      33       23            33          63           23             (3 3             3
                                                                                          2)( 3 32              3
                                                                                                                    6            22 )


                             (3 3             3
                                                  2)( 3 32                   3
                                                                                 3    3
                                                                                           2         3
                                                                                                         22 )           ( 3 3)3 ( 3 2)3              5.
                                                                                                                                                           59




                                         1                                  10                                  1                                  2
                                                     ;                              ;                           3
                                                                                                                        ;                      5
                                                                                                                                                       .
                                             3                                  5                                   3                              7


    1           1               3            3                 3
                                                                 ;
    3               3           3            2
                                             3                3

10              5       10 5                 10 5
                                                                     2 5;
    5           5                5   2
                                               5
                        3
1               1               32       3
                                             9               3
                                                                  9
3           3
                                                                    ;
    3               3   3
                                32       3
                                             3   3
                                                                 3
                            5                        5
    2               2           74       2               74          2 5 74
5               5
                                                                            .
    7               7       5
                                74               5
                                                     75                 7

                                                 1                                  3            2                                       1
                                                             ;                                     ;                                           .
                                             7           2                          3            2                                   2 3 1


        1                       1                7        2                     7        2                          7        2               7 2
                                                                                                                        2        2
                                                                                                                                                 .
    7 2                     7 2                  7        2          ( 7 2)( 7                    2)        ( 7)             2                3

3           2           3            2 3                 2                (3            2) 2             9 6 2 2                      11 6 2
                                                                                                            2                    2
                                                                                                                                             .
3           2           3            2 3                 2           (3        2)(3              2)         3       ( 2)                 7

  1                       1   2 3 1                                        2 3 1                     2 3 1                  2 3 1
                                                                                                                                  .
2 3 1                   2 3 1 2 3 1                                       (2 3) 2 1                   12 1                    11

                                         12                        7                    4                 1
                                                                                                               .
                                         5 1                      5 2          3             5         11 5 14


        12                       7                       4                 1
        5 1                     5 2              3               5      11 5 14

                            12                   5 1                     7               5       2              4     3              5          1
                            5 1                  5 1                    5 2              5       2      3           5 3              5       11 5 14
60

                           12 5 12                   7 5 14                        12 4 5                      1
                             5 1                      5 4                            9 5                    11 5 14

                           12( 5 1)                                                                            1
                                                     7 5 14 (3                                     5)
                               4                                                                            11 5 14

                                                                                                  1
                          (3 5          3 7 5 14 3                                     5)
                                                                                               11 5 14
                                                       1
                          (11 5 14)                         1.
                                                    11 5 14
                                               x    y                x     y                                                  2x 1                  2x 1
                                               1        1            1         1
                                                                                   ;                              1       2            1
                                                                                                                                                          .
                                           x   2
                                                    y   2
                                                                 x   2
                                                                           y   2                                  3       3            3           1 3 2x
                                                                                                                4 x            (2 x)           1


         x       y              x      y             x       y                 x           y
         1           1          1          1

     x   2
                 y   2
                            x   2
                                       y   2        x            y             x               y

                 (x         y )( x                 y ) (x                y )( x                y)
                                     ( x            y )( x                 y)

                 x x                 x y           y x       y y (x x                              x y          y x           y y)
                                                     ( x                  y )( x                   y)

                 x x                 x y           y x           y y           x x                 x y       y x              y y
                                                    ( x                  y )( x                    y)

                          2x y                 2y x                       2 x y( x                           y)               2 xy
                                                                                                                                           .
                 ( x                 y )( x                 y)           ( x               y )( x               y)            x        y
                 2x 1                           2x 1                                   2x 1                           1 2x
         1   2                   1                                           1         2                1
         3   3                   3             1 3 2x                      2 3         3                3
                                                                                                                      3
                                                                                                                       2x 1
     4 x                 (2 x)        1                                  (2 ) x                (2 x)        1
                                2x 1                         1 2x
                            2                  1                     1
                            3                  3
                 (2 x)               (2 x)         1 (2 x) 3 1
                                                                                                         61

                   1                                 2               1

(2 x 1) (2 x) 3 1          (1 2 x) (2 x) 3                   (2 x) 3 1
                   1              2              1

               (2 x) 3 1    (2 x) 3       (2 x) 3 1

           1                 1               2                   1                    2              1

(2 x)(2 x) 3     2x    (2 x) 3   1 (2 x) 3               (2 x) 3         1 (2 x)(2 x) 3   (2 x)(2 x) 3    2x
                                                         1   3
                                                         3
                                             (2 x)                   1
     2                      1

(2 x) 3 (1 2 x) 2(2 x) 3         4x
                                      .
            2x 1
62




                                                                                                                        0                                               0
      0                                    0                                   0                                1                                               2
     2;                                   3;                                   1;                                           ;                                               ;
                                                                                                                4                                              3
            0                                              0                                                                    0                                               0
                                           2 3                                     0                            x 1                                    x                2
                ;                                              ;               x ;                                  ;                                                               .
      3                                     3                                                                    2

     ( x0       2 y )0 ;                               (x 2z0                  3 y 0 )0 ;                                       x0      y0        ( z 2 )0 .
                                                                                                                                                   0
                        2 0                                        2           0 0                                                  x      y
     2(a b ) ;                                         (2(a                b) ) ;                                                                          .
                                                                                                                                    3     40
                                                                   3
     x 0 y 0 1;                                        x0            ;                                                          x0 : x2 ;
                                                                   4
                                                                                                                                                 1
     x0 x2 ;                                           x2 : x0 ;                                                            x2 x0                  .
                                                                                                                                                 2
     ( x 3 y )0 ;                              x0 3 y 0 ;                              x0       (3 y )0 ;                               x 0 30 y.
     2 1;                                      2 2;                                    3 2;                         4 3;                           ( 2) 2 .
     1                                    3                                    2                                5                            a
        ;                                    ;                                    ;                                 ;                           .
     23                                   24                                   32                           (_ 4) 1                          a2
                1                                      2                                    3
      1                                        1                                       3
                    ;                      1               ;                    1               ;           (0.4) 1 ;                     ( 0.5) 1 .
      2                                        2                                       4
                                                                                                                                                                2
                                                                                                                                          1 2
     3 2 1;                               22 3 1 ;                             24 4 2 ;                     82 4 4 ;                                                .
                                                                                                                                          3 3
                                                                                       x                                        3( x y )
            x 2;                               x 4;                                       ;                                              .
                                                                                       y3                                       ( x y) 2
                                                                                                    1               2                    3
                2             2       3            2                                       1                1                      1
            2            3        1            4 ;                                                                               1           ;
                                                                                           2                3                      3
                          1                    2                       3                                4                       3                      2
                    1                 1                        1                                1                   1                        1
                                                                           ;                                                                               .
                    3                 3                        3                                2                   2                        2
                                                                                                                                                  63

                                                                                                                                    3     0
              2                  3                   1                            5                     2                      1
(( 2)                 ( 3) ) : ( 4) ;                                         (2 ( 5) ) 25                             2                      ;
                                                                                                                               2
                         0                                        2
  1                 2                      1           1
1                 1                      1           1                    ;
  2                 3                      3           2
             2                       3       2                        2               2
     1                    2                                   1               3
                        1                                 1                                   .
     3                    3                                   3               4



ax 1 y 2 ;                                               (a 2 ) 3 x 2 c 3 ;                                         (ab 2 c 3 ) 2 x 4 .
         2                                                 1                                                             3
               ;                                                ;                                                                             .
x    2
         33 a4                                           x y 4z
                                                              3                                                        1
                                                                                                                    4 x (x 1
                                                                                                                                    y)    2



x 3y 2z                                                  x 1 y 3 ( z 2) 2                                                a 2b
        ;                                                                 ;                                                         .
a 2b 3                                                    3 1 x( y 2) 1                                             (b 1) 3 (a 1) 2

                  3              1                          ( x z) 2                                                x 3 a 4b 5
(x       y)                                  ;                       ;                                                         .
                        2    1
                                     x   3
                                                         ( x y z) 2                                                  3 1y 2

                                                          2
a                                                    x                                            1                                1
   ;                                                          ;                                     ;                                 .
b2                                                   y                                            x                                a3
                                                                                                            3
xy                                               xy 2                                              ab                               ax
   ;                                                    ;                                                       ;                          .
z2                                               z 3t 4                                            xyz                             (by ) 2
         x                               (x          y) 2                                  x                                       x2
                  2
                    ;                                     ;                                      ;                                       .
(x           y)                          (x          y) 3                              2
                                                                                      y ( x y) 3                               x( x a )3
   3x 2 y 3                                ( x 1) 3                                   2x 3 y                         2( x y ) 2
             ;                                        ;                                       ;                                  .
4 z 2 ( y 1)                             4 2 ( x 2) 5                                 3x 3t 4                       4a ( x y ) 3
                                                           n 1
 x                                               1                                     x                             am
     ;                                                            ;                        ;                               .
y 2n                                         x       y                                y zct
                                                                                                                    bn c t
64
                                                                                       1
                    2                   1                                      2            12
                2           31             ;                                                   ;
                                        43                                     5            23
                                    3                                                                  2
                 1    3                                                            3           1
                  2
                    :                   ;                                      2           1                   (2 1 ) 2 .
                3     5                                                                        4


                        2                                                                                                                    2
         x3 x 2                 x3 : x 2                                                                                      x2
                                         .                                             (0.2 x 3 y 2 ) 2                                          .
           x4                      x                                                                                          2 y3
                        3                           2                                                          2                         2
           7 x2                  9 y2                                                      x3 y 2                       x2 y 3
                                                        .                                                                                    .
           3y4                   49 x 4                                                     9 y2                         6 y3
                                                                                                       2                        3
          8 x 2 y xy     15 x 2 y 2                                                        x5                          2 y2
                 : 3 2 :            .                                                                                                .
         12 y 2 x x y     3xy 1                                                            y2                          x4
                    5                       2                                                                      2                 2                     4
           x3               3y7                                                            1 1 3                          x2                         2x2
                                                .                                            x y                                                               .
          y5                 x3                                                            6                              y2                          y3
                            3
         1 1 3
           x y                  : ( x 2 : y 8 ).
         2
                                                                                                   11
     ( x 2 y 3 ) 8 ( x 4 y 6 )6 (( x 2 y ) 4 : ( x 5 y 3 ) 2 ) 1
                                                              :                                            .
     ( x 4 y 6 ) 2 ( x 1 y 2 )3        ( x3 ) 5 y 7             y


                                                                                                   5
                3
                    8 27;                                     3
                                                                  2   3
                                                                          4;                           65 35 ;
                                                              4
                    20           5;                               72 18;                               27              12.
     5
         73 22          5
                            7 2 23 .                                                       8 4 3 64 .

         5( 2 1) 2                  3 (1                    2) 2 .                         12                  45         18.

           1        1                   5
     5                20                  20.                                          (3 5 2)(3 5 1).
           5        2                   4
                                                        1
         2 6 4 3 5 2                                      8           3 6.
                                                        4
                                                                                       65


(2 6 3 5 1) : 3 ( 6                2 5 3) : 5 (3 6               4 5 1) :15.

    1
      8 3 2            10   ( 2 3 1.6 5 0.4).
    2

(12 50 8 200 7 450) : 10.

    1 1         2 1    2 4   2 1
                           :     .
    3 2         3 3    7 5   7 8

( 4 2 3               4 2 3 )( 4 2 3                  4 2 3 ).


2 18 3 8 3 32                50.

(2 8 3 5 7 2)( 72                     20 4 2).

(4          6)(3 2 5 3).


        1                         3                                       2
            ;                 3
                                      ;                               4
                                                                               ;
        3                         2                                       49

    2                                     1                                    2
3
                                                  .                                .
    3                                 2       3                           2 2 1
        1                                 1                                    4
                 .                                .                                .
2 2 3 3                               2       3                           3 2 1
        1
                 .
2 2 3 3
                                         ax 2       bx c          0
a , b, c                                             a        0




                   b    0,
                                        ax 2    c    0, a         0.




                       2 x 2 18 0;                                          9 x 2 16 0;
                       3x 2    7   0;                                        x 2 1 0.



                                                    b)
2 x 2 18 0                                               9 x 2 16 0
2 x2       18                                            9 x2         16
x2      9                                                         16
                                                         x2
                                                                   9
x              9
                                                                            16
x       3.                                               x2
                                                                             9
    x      3       x      3.
                                                                      4
                                                         x              .
                                                                      3
                                                                                                                 67



 c) 3x 2 7 0                                                 d) x 2 1 0
    3x 2         7                                                x2       1
                 7
     x2
                 3
                     7
     x                 .
                     3




  4 x 2 36 0.                                          4 x 2 81 0.                     36 x 2 100 0.
  2 x 2 10 0.                                          x2    2 0.                      16 x 2 9 0.
  16 x 2         25 0.                                 25 x 2 36 0.                    25 x 2 81 0.


                                c       0,
                                                      ax 2   bx   0, a     0.



                            a) 3 x 2 5 x 0;                       b) x 2        x 0;     c)       3x 2     2x    0.
Zgjidhja.


    3x 2         5x        0                                                            x2    x    0
     x(3 x 5) 0                                                                         x( x 1) 0
     x       0             3x 5 0                                                       x 0              x 1 0
                                        5                                               x 0              x 1.
    3 x 5 0, x                            .
                                        3
                                         5
         x       0              x
                                         3


        3x 2           2x           0
    x( 3x                  2) 0

    x        0                 3x         2       0
                                          2
     x       0              x                 .
                                          3
68




      x2        x     0.                     x2      2x     0.                                  2 x2             3x    0.
           2                                     2                                                  2
      4x         5x        0.                x       ( a b) x            0.                    ax           2x    0.
                7                                                                                       x
      x2          x        0.                5x2      4x        0.                         x2                0.
                2                                                                                       2




                                x 2 5 x 6 0;                                  x2          3x    2 0;
                            c) x 2 6 x 9 0.
Zgjidhja.

     x2 5x 6 0
     (x        2)( x 3) 0
               x 2 0                 x 3 0                                    x       2             x 3.


                                                                         x2       6x 9 0
          2                                                              2
      x         3x 2 0                                     ( x 3)
      ( x 1)( x 2) 0                                       x2        6x 9 0
      x 1 0                     x 2 0                      ( x 3) 2               0       ( x 3) 0
      x 1                  x    2.                         x 3.
                                                                     x        3
                                                                                                                            69




 x2       3x         2    0.                          x2       3x 4         0.                    x2     x 6      0.
 x2       4x 3            0.                         x2        5 x 14       0.                    x2     2 x 15        0.




       ax 2    bx c           0, a        0

                                                           b     b 2 4ac
                                              x1,2                       .
                                                                 2a
                D        b2    4ac




                                                b          D            b            D
                                     x1                        , x2                      .
                                                     2a                     2a


                                                                 b
                                                           x       .
                                                                2a




                                                     x 2 3 x 4 0.


                a 1, b             3, c          4.


              ( 3)        ( 3) 2      4 1( 4)              3      9 16           3           25        3 5
x1,2                                                                                                       .
                           21                                     2                      2              2
70

                 3 5      8                   3 5      2
       x1                       4, x2                       1.
                  2       2                    2      2


                                                x 2 8 x 16 0.


                      a 1, b 8, c 16.


                  8      ( 8) 2        4 16     8     64 64          8 0
     x1/ 2
                           2                          2               2
                         8
       x1        x2               4.
                         2



                                                 x2    x 1 0.


             a 1, b          1, c 1.


                  ( 1)        ( 1) 2     41     1     1 4        1       3
     x1/ 2                                                                   .
                              21                      2              2
             D        3 0,




                             ( x 2) 2     (2 x 3) 2    13 4 x.


     ( x 2) 2         (2 x 3) 2 13 4 x
     x2      4x 4 4x2             2 2 x 3 9 13 4 x
     5x2         8 x 13 13 4 x
                                                                                                        71


       5 x 2 12 x      0
       x(5 x 12) 0
                                                                    12
                                            x1       0    x2           .
                                                                     5
                           (2 x 15)(2 x 7) ( x 36)( x 8) 36 0.


       (2 x 15)(2 x 7) ( x 36)( x 8) 36 0
       4 x 2 14 x 30 x 105 ( x 2 8 x 36 x 288) 36 0
       3x 2      44 x 105 44 x 288 36 0
            2
       3x        141 288 0 3 x 2            147
       x2       49         x1,2       7.


                                                           5   1
                                                     x2      x             0.
                                                           6   6




                                     5           1
                      a 1, b           ,c          .
                                     6           6
                               2
                5          5                     1             5     25         24   5       49   5 7
                                    41
                6          6                     6             6     36         36   6       36   6 6
x1,2
                               21                                   21                   2         2
                5 7    2                         5 7               12
                6 6    6          1              6 6               6
x1                                  , x2                                   1.
                 2     2          6               2                2




                                                           5   1
                                                     x2      x             0
                                                           6   6
72

           6 x2    5 x 1 0.
                          a     6, b 5, c     1.

       5          52 4 6 ( 1)         5      25 24        5        49     5 7
x1,2                                                                          .
                    2 6                     12                12          12
       5 7         2   1              5 7     12
x1                       , x2                        1.
       12         12   6              12     12




1.                              2.                                  3.


4.                              5.                                  6.


7.                              8.                                  9.


10.                             11.                                 12.

13.                             14.                                 15.
16.                             17.                                 18.
19.                             20.                                 21.
22.                             23.                                 24.
25.                             26.                                 27.
                              y   ax b a, b       R   f ( x)   ax b


                                                                    f ( ) 0.
                                                                    b
        f ( x)        0                    ax b   0        x          ,a   0.
                                                                    a
                                                                                b
                                                                                  ,
                                                                                a
    b                     b
f           a                 b   0.
    a                     a

                 b
        A          ,0
                 a
                     Ox.
        b       0,                     y    b.             (0, b)


            a        0,



                              y   x;                   y       x.
74


                                  y       x

            O(0, 0); A(1,1)                       y           x.




                                  y       x
     A(0, 0), B (1, 1)

                  y                                       y
                          y   x               y       x

                  1
                                                                   1
                      0   1           x                   0                  x
                                                          -1




                          1                       1
                      y     x;                y     x 2.
                          2                       2




                                      1           1
                                  y     x
                                      2           2
                                                                                   1
                                                                   A(0, 0), B 1,
                                                                                   2
                      1
         x 0; y         0 2 2
                      2
                1
     x   2; y     2 2 3.
                2
                                                                          75




                               1
                           y     x 2
                               2
     A(0, 2), B (2, 3)
                                                                               y
                                                    y         1
         y                                              y       x 2
                                                              2
                         1
                     y     x                        3
                         2                          2
         1
                                                    1
         2                                                                     -1
             0       1                              0   1 2           x        -2
                                                                                    -




                     1
                 y     x 3;         y    x 2.
                     2




                               1         3
                           y     x 3            2
                               2
A(0, 3), B (2, 2)




                               y   x 2
A(0, 2), B (1,1)
          76
               y                                                              y

                                                      y      x 2


                         1 2                                                  2
                                                 x                            1
               -1
1 2   x                              1                                        0   1 2           x
               -2              y       x 3
                                     2
                    -3




                                                                    1
                          y        2 x 3;                y     3x     .
                                                                    3




                                            y        2x 3
                                                          A(0, 3), B (1,1)




                                                     1        1           8
                                     y          3x
                                                     3        3           3

                    y
                                                                                  y

                    3     y        2x 3
                    2                                                             1
                               3                                                        1
                    1                                                             3                  x
                               2                                                  -1
                    0     1                     x
                                                                                  8
                                                                                  3                  1
                                                                                            y   3x
                                                                                                     3
                                                                     77




y   2.          y        3.        x       4.        x     4.
y   x 1.        y       x 3.       y        x 4.     y     x 3.
      1                 1
y          x.       y     x 2.         x    y   4.       x y    7.
      2                 3
                    2     1
    x 2 y 3.          x     y 5.
                    3     4
                       1    2
                  x           2 x.
                       3    3


     1        2
x               2x
     3        3
              2   1
x 2x
              3   3
    x 1| ( 1)
x        1.
              x   1                          x (   , 1).




                      x (   , 1)

                                     1   0                 x


                        x    1
                                                                                           79


                          2( x 1)        x 4.


2( x 1)       x 4
2x 2          x 4
x 6,
                                                               x (      ,6].

                                         x (               , 6]

                                                     0                         6       x


                                     x       6
                                    1                    1
                          3 x                x             .
                                    2                    2


          1                1
3 x                   x
          2                2
      3               1
3x            x
      2               2
                  1       3
3x x
                  2       2
2x        2
x     1,
                                                               x [ 1, ).

                                                               x [ 1,    )

                                    -1           0                                 x




                                x        1
80




                  2(3 x) 3(1 2 x).


     2(3 x) 3(1 2 x)
     6 2x 3 6x
         2x 6x 3 6
     4x      3
            3                      3
     x        ,    x          ,      .
            4                      4
                  3               1
                    ( x 1)          ( x 3);
                  2               2


     3         1
       ( x 1)    ( x 3) | 2
     2         2
     3( x 1) ( x 3)
     3x 3 x 3
     2x 6
     x 3.



                                  2 x 3 3x 5   x   2x 1
                                                        .
                                     4    6    2     3



     2 x 3 3x 5        x   2x 1
                                12
        4    6         2     3
     3(2 x 3) 2(3 x 5) 6 x 4(2 x 1)
     6 x 9 6 x 10 6 x 8 x 4
                                                                       81

    2x       5
            5
    x         .
            2
                                                               5
                                                           x     ,      .
                                                               2




                       -5/2 -2              0                      x




                  x     2.



                                    2x 1        x 1 1
                                         1            x.
                                     10          10 2




    2x 1          x 1        1
         1                     x | 10
     10            10        2
    2 x 1 10          x 1 5x
    6 x 12
    x 2.

(   , 2).



                                        0   1    2             x
82


                 x 1.



           A B        0         (A 0    B       0) ( A 0           B    0)
           A B 0                ( A 0 B 0) ( A 0 B 0)
           A
                0         ( A 0 B 0) ( A 0 B 0)
           B
           A
                0         ( A 0 B 0) ( A 0 B 0).
           B



                      ( x 2)( x 3) 0.


( x 2)( x 3) 0              (x 2 0          x 3 0) ( x 2 0                      x 3 0)
     (x     2 x 3) ( x                 2 x 3)            x 3 x          2
     x (       , 2)       (3,     ).


                                            2             3                                       x



                      ( x 1)( x 4) 0.


( x 1)( x 4) 0              (x 1 0       x 4 0) ( x 1 0                      x 4 0)
     (x        1 x         4) ( x       1 x         4)        (x       1 x      4)       x [ 1, 4].


                                        -1          0                       4                   x
                                                                                         x    1 x     4,
                                                x       1, x 4
                                                                                           83


                    x 1    3
                             ;
                    x 2    2




     x 1 3      x 1 3       2( x 1) 3( x 2)
                      0                     0
     x 2 2      x 2 2            2( x 2)
       2 x 2 3x 6          x 4           4 x
                    0              0| 2       0.
          2( x 2)       2( x 2)          x 2



(4 x 0 x 2 0) (4 x 0 x 2 0)                       (4       x        x     2) (4   x   x   2)
          (x      4 x   2) ( x   4 x   2)    (x    4 x              2).
      x (2, 4].


                           2                 4                                        x




                                       x 1
                                  1          4;
                                       x 3



 x    1         x 1       x 1 x 3
          1         1 0              0
 x    3         x 3         x 3
            ~           ~
 x    1         x 1       x 1 4 x 12
          4         4 0                0
 x    3         x 3          x 3
        2
              0               (1)
      x 3
       3 x 11
                 0            (2)
        x 3
                                        x 3 0          x       3.
84

         ( 3 x 11 0 x 3) ( 3 x 11 0 x 3 0)
            (3 x 11 x 3) (3 x 11 x 3)
                        11                       11                                      11
                    x          x 3           x             x   3             x 3 x          .
                         3                        3                                       3
                        11
      x (      ,3)         ,     .
                         3




                                     3                11                       11               x
                                                                         x        ,
                                                       3                        3



                                                 11
                                         x          ,
                                                  3




                                                 x 3 y 6.


                                 x 3y            6.
                    x   0                                            y        2.
                                                               (0, 2).
                    y   0,               x 6.
                                                               (6, 0).
                         x 3y        6                                       (0, 2),(6,0).


P ( x, y )


         P (0,0).
                                                                                     85

                                           0 3 0 4,
               P(0,0)




                                                                     x 3y   6.




                          y

                                                  pika "testuese"
             x+3y=6


                         2

             x+3y<6 1

                   (0,0)              1      2         3     4   5      6        x




                        P ( x, y )

                                     P (0, 0).
                                                 x 3y        6
               x 3y        6,
                                     x 3y        6.

Shembulli 2.
                                            3x        y 5.




        3x     y   5.
86



     x    0         y 5.
                           5
     y    0         x        .
                           3
                                    5
                   (0,5)              ,0 .
                                    3
                                                     3x    y       5,
                           (0,0).
         3 0 0 5


                                             y



                                             5
                                             4
                                             3   3x+y=5
                                             2
                                             1

                                       (0,0)     1   2         x



                                                                   2x 3y 9 0


                                                                        2x 3y 9 0
                                 9
                     (0, 3),       ,0 .
                                 2
                                                         2x 3y 9 0



          21 31 9 0
                                                                               87



2 x 3 y 9 0,


                                   y
                                               (1,1)
                                   1

                                   5       1      2    3    4   5      x
                                  -1
                                  -2
                                  -3




                    x 3                    y      1.

                              x    3.

        x 3,                                                    x 3.
                              y        1


         y     1,                                           y   1.


                y
                                                       y



                                   x=3



                    0   1 2       3        x            0                  x
                                                       -1         -1
                                                                 y=
88



                                             x        y 1
                                             x        y   2



x    y 3          x    y   2


     y                             y                                       y




                                   2                                       2
                                                                                            x-y=1
                      x-y=1        1                                       1

(0,0)         1                x (0,0)       1 2                   x   (0,0)        1   2       x
         -1                                   x+y=2                            -1
                                                                                        x+y=2




                                             x    y       2
                                             2x       y   4


                                  x      y        2           2x   y   4
                                                              89

                         y




                                           2x-y=4
     x+y=-2

                   -2    0       1 2            x
                        -2
                        -4




                    x 2y 3
                    3x 4 y 0
                    x 3


              y



                                                    x=3


x+4y=0
              0              1         2        3         x

              -1
              -2
 x-2y=3
90



                  x   y 1
                  x   y 3
                  x 2y 3 0



             y



             2
     x+y=1        3
             1

             0        1     2   3   4   5   x
             -1

             -2

             -3
                                                                                                            91




                                                 3( x 5)      x 3                  2x 1       x 1   x
      2( x 4)     x 1.                                            .                               1   .
                                                     4         5                     3         4    6
      x 1 2x            1                        x 1    x 1         2              x 2
                          .                                           .                      0.
       4  3            12                         3      5          3              3 x
      x 1                                        2x 5      1                       3 x
          0.                                                 .                                1.
      x 1                                         x 5      2                       1 x
       4x 1                                              x 1                             x 1
            1.                                     2                    2.         0                   3.
       2x 3                                             2x 3                             4 x
        8 x 3( x 2)               x 2.                   3 x 3 5( x 1) 2.

                              1
       ( x 8)     2 x                 7.                 7 x 1 16( x 1) 2.
                              2

                                  1                        x 5
       (8 x)           7 x             2 x.                             0.
                                  7                        x 9
       5 x 1.5                                    5x                                              1
                       0.                                    0.                    x     3              .
        x 49                                     3x 1                                         x 1
        2x                                        x 10                              x 17
                  0                                          0.                                   0.
       18 x                                        x 5                               8x
       x 1       3                               5 2x        1                     6 x
                   .                                           .                                  2.
       x 2       2                                5 x        2                     3 x
        x 1                                      (2 x 1) 2                         ( x 1) 2
                 0.                                                0.                              0.
       x2 3                                        x 5                               x 2



31.                                        32.                               33.

34.                                        35.                               36.
37.                                        38.                               39.
92




40.   41.   42.



43.   44.   45.




46.   47.   48.
                         1
                         4
         a ) 160;       b) 240;      c) 324;      d ) 404.




b) 60;         c) 81;             d ) 101.


             a ) 100;     b) 120;       c) 360;      d ) 600.




b) 72;         c) 216;               d ) 360.
94



                         1                     1                     1                    1
                    a)     ;              b)     ;             c)      ;           d)       ;
                         2                     4                     5                    8
                         1                     1                     1                           1
                    e)     ;              f)      ;             g)      ;                 h)        ;
                        10                     20                    50                         100
                    i ) 4;                j ) 6.




                                         1     100
a)          1 100%,                                % 50%;
                                         2      2


b) 25%;             c) 20%;            d ) 12.5%;              e) 10%;
f ) 5%;                  g ) 2%;               h) 1%;                      i ) 400%;
j ) 600%.


              a ) 13%;           b) 21%;             c) 29%; d ) 31%;
              e) 70%;            f ) 92%;               g ) 100%;          h) 124%.


             13                21                     29                    31
a ) 13%         ;        b)       ;            c)        ;           d)        ;
            100               100                    100                   100
             70      7 10        7                                  92      4 23        23
e) 70%                             ;                 f ) 92%                               ;
            100     10 10       10                                 100      4 25        25
             100                                              124        4 31      31
g ) 100%         1;                            h) 124%                                .
             100                                              100        4 25      25

            a ) 24%;       b) 19%;                           c) 37%;                            d ) 75%;
     e) 125%;        f ) 115%;                                  g ) 150%;                       h) 305%.


             24
a ) 24%             0.24;              b) 0.19;                c) 0.37;            d ) 0.75;
            100
e) 1.25;                         f ) 1.15;              g ) 1.5;           h) 3.05.
                                                                          95



                  7%
             a) 77;           b) 194;        c) 208;        d ) 215.


                                                7        539
a ) 7%                                              77            5.39;
                                               100       100
b) 13.58;                c) 14.56;                 d ) 15.05.
                  12%
             a) 96;           b) 126;        c) 512;        d ) 1000.


a ) 11.52;            b) 15.12;              c) 61.44;         d ) 120.
                  125%
             a) 25;           b) 150;        c) 270;        d ) 305.


a ) 31.25;        b) 187.5;            c) 337.5;         d ) 381.25.


                  0.5%
             a) 72;           b) 160;        c) 204;        d ) 300.


                                              0.5
a ) 0.5%                                           72 0.36;
                                              100
b) 0.8;               c) 1.02;                 d ) 1.5.
                    12.6%
             a) 90;           b) 126;        c) 192;        d ) 790.


                                               12.6
a ) 12.6%    90                                      90 11.34;
                                               100
b) 15.876;            c) 24.192;                d ) 99.54.
                    125.2%
             a) 74;           b) 92;         c) 1000; d ) 12520.
96


a ) 92.648;           b) 115,184;          c) 1252;      d ) 15675.04.
                      6%
              a ) 66.6         b) 72.8;    c) 92.9;      d ) 100.6.


a ) 3.996;         b) 4.368;         c) 5.574;        d ) 6.036.




                                                              31
             31%                                                 800      248
                                                             100

                                                                          27
                                                                             800   216
                                                                         100

       248 216       464


                                                 800 464 336




        31%                                                 27%
                           31% 27% 58%
                   42%
                                               97



                                     21
      21%                               800 168.
                                    100




                   30
30%                   140   42,
                  100
                                       45
            45%                           140 63,
                                      100
                                  140 (42 63) 35
98




                                                         3%            20%




                          2                          1
                     a)                       b) 1
                          5                          3

                                  2
a)                                  x   75.      2 x 5 75,
                                  5
            375               2
       x         187.5.
             2                5
b) x       49.5.
                     a) 0.5                   b) 1.6


                                                              80
a)                                0.5 x 80.              x        ,   x 160.
                                                              0.5


b) x       46.25.
                          1                                           9
                     a)                       b) 0.7                    .
                          3                                           7


a ) x 9.9;
           90
b) x          .
           49
                                                                           99


                     a ) 45%                         b) 27%




                                          45
a)                                           x 90.        45 x   90 100,
                                         100
                45 2 100
            x               200.              45%
                   45
        500
b) x        .
         3
                     a ) 0.7%                        b) 3.6%


a) x   70;                      b) x   2500.
                     a ) 42%
                b) 104.5%


a) x    260;                    b) x   250.
                     42%


     100 78     1300
x                    .
       42         7
                            36%




x    200.
100




   24                           100 x 24 x
x     x     76,                              76.        76 x    76 100
  100                              100
x 100.


                                                                     a 12 cm
                  b 15cm c 15cm       d 18cm.




P     a b c d       12 15 15 18 60cm.                          a 12cm
1                                                   1
5                                                   5
                    20%.
                                                                           1
          b 15cm                                   c 15cm
                                                                           4
              1                  1
                60 15).                                                  25%.
              4                  4
                            3                            3
        d    18cm
                           10                           10
                    30%.

				
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