GR 10 WISKUNDE TUTORIAAL: EKSPONENTE

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					GR 10 WISKUNDE TUTORIAAL: EKSPONENTE

10.1.2: Eksponente, Wortelvorme en Afronding.

         Toepassing van eksponentwette om uitdrukkings met heeltallige magte te
          vereenvoudig.
         Om die plekwaarde van 'n wortelvorm in terme van twee heelgetalle te
          bepaal.
         Afronding van rasionale en irrasionale getalle.

Opsomming van eksponente wette:

           a m  a n  a mn
           am
             n
                a mn
           a
                      an
           a 0  a nn    1
                      an
            n   0n    a0    1
           a a  n  n
                        a     a
           (ab)  a b
               n    n n

                n
           a    an
                n
           b    b
           (a )  a mn
             m n




1.        Gebruik eksponentwette om die onderstaande uitdrukkings te vereenvoudig:

1.1       am x an  ap                              1.2    [ ( am )n ]p

                                                                                         2 3.2 4
1.3       [( am )n ]p                               1.4    2-3 + ( a + b )0 x
                                                                                           2 5
           a 2   3
1.5        0
           2b                                     1.6    [ ( a -2 ) -3 ] –5
                

          2a 3b 2  8a 5 b 10                              2 x   3
                                                                         y 2   3x 1 y 3 
                                                                              3                     2


                                                                          6 x        y2 
1.7                                                 1.8
               2a  2 b 7                                                         2      3
                 1
       (a  x) . a 2  x 2
                 2
1.9                     1
                                    gee die antwoord met positiewe eksponente.
             (a  x)    2




        3 p q   3 p
                 2   1 3           1
                                         q2 
                                            2


                pq 
1.10                                             gee die antwoord met positiewe eksponente.
                             2 4



2.

Probeer die volgende

Vereenvoudig die volgende saamgeselde algebraïse breuke so ver moontlik:
                6
       5n  7 
1.              n
           3 8
       5 2 
          n     n

             1
2.
                 1
       1
                   1
            1
                 x 1

3.     Beskou die volgende klein vierkante met sye x cm, wat saamgevoeg is om
       diagramme M en N te vorm.
3.1    Watter diagram het ‘n area van 3x 2 ?                                                       3x
3.2    Watter diagram het ‘n area van 3x 2 ?
3.3    Dink jy dat 3x 2  3x 2 ? Verduidelik.                  3x                           3x
3.4    Wat is die area van M en N as x = 1, 2 en 3?
                                                                       x
                                                                              M                    N
4.     Los op vir x sonder die gebruik van 'n sakrekenaar:
4.1    25x+1 = 125
4.2    82x = 32x+1
       8 x 1
4.3        x
                 16 1
        4
4.4    10 ( x  4)( x  2)  0,00001

5.     Jy is die onderwyser van die nuwe graad 10 Wiskunde klas. Skryf ‘n opsomming
       van die reëls en definisies wat benodig word vir die vereenvoudiging van
       eksponente. Gee ‘n voorbeeld in elke geval, sodat enige nuwe, of onervare
       onderwyser dit kan aanbied indien jy ‘n konferensie moet gaan bywoon.

6.     Verduidelik die onderskeid tussen ‘n vierkantswortel en ‘n wortelvorm.
                           7.      al’Khwarizmi, wie se volle naam Abu Abd-Allah ibn
                           Musa al’Khwarizmi is, is gebore rondom 790 n.C naby
                           Baghdad, en het gesterf rondom 850 n.C. Sy belangrikste
                           bydrae geskryf in 830 n.C, was Hisab al-jabr w’al-muqabala.
                           Na aanleiding van die al-jabr in die titel kry ons algebra wat
                           sinoniem is met die wetenskap van vergelykings. Sy geskrifte
                           het gely tot die ontdekking van ‘n metode en ‘n formule vir die
                           benadering van die vierkantswortel vir enige getal x. Die
                                                 x  p2
                           formule is x  p            waar p die vierkantswortel vir die
                                                 2p 1
naaste vierkant kleiner as x is.

Voorbeeld: Gebruik bostaande formule om die vierkantswortel van 19 te benader.
Aangesien 16 die vierkant is kleiner as 19, dan is p  16  4 .
           x  p2
     x  p
           2p 1
           19  16
  19  4 
           2(4)  1
            3
       4
            9
       4,333
Let op dat die antwoord nie veel verskil van die sakrekenaar se antwoord wat 4,359 is
nie.

7.1      Gebruik nou die formule om die volgende te benader:
         a).    27                                       b).      3 67

7.2      Kontroleer jou antwoorde in 7.1 deur die gebruik van die sakrekenaar.

8.       Gegee die getalle:     56 ;     132 ;     379 ;     4000
         (a) Skat waar hierdie getalle sal lê op ‘n getallelyn. Regverdig jou antwoorde
             sonder die gebruik van ‘n sakrekenaar.
         (b) Bepaal nou die waarde van elkeen korrek tot twee desimale syfers met behulp
             van ‘n sakrekenaar.


9.       Sonder om die sakrekenaar te gebruik, bepaal waar die antwoord van die
         volgende uitdrukking op die getallelyn sal pas.
         1     3      3 1 2 
            of 6 1     3  1 1
                   2
         2     4      2 3 3  2
                                                          x               183
10.    Mev. Naicker, die eienaar van ‘n slaghuis in
       Knysna, wil die area van die bestaande
       reghoekige parkeerarea verdubbel. Tans is die                     16 836 m 2       92
       afmetings 183 m by 92 m. Vir die parkeerarea
       om te verdubbel, moet beide die lengte en
       breedte met x meter vergroot word. Bepaal die
                                                                                              x
       afmetings van die nuwe parkeerarea.


                                                                   A
11.   Sipho moes ‘n driehoekige teël ontwerp
      waarvan die sye ewe lank is.
11.1 Bepaal die waarde van x as AB AC                                           3x  2
11.2. Kan hierdie driehoek gelyksydig wees?
      Verduidelik.                                      5x  8

                                                                                          C

                                                         B                x 9


12.    ’n Maatskappy het die volgende produksiekoste funksie (C) en inkomste funksie
       (R) wat hulle gebruik om x kopieë van ’n nuwe jeugtydskrif te vervaardig en te
       verkoop:
       C(x)  10x  600
       R(x)  80x  x2
       Bereken hoeveel kopieë van die tydskrif verkoop moet word sodat die
       produksiekoste en inkomste gelyk sal wees.

13.     Om die wins wat hy op ’n volvrag van sy trok maak, te bereken, gebruik ’n
                                                3
       vervoerkontrakteur die formule: P   x 2  12x , waar P die wins (in rand) per
                                                20
       uur is en x, die spoed waarteen hy ry (in km/h).

13.1   Wat is sy spoed wanneer sy wins R180 is?
13.2   Wat is sy maksimum wins?
13.3   Hoe vinnig moet hy ry om sy maksimum wins te bereik?
13.4   Wat is sy spoed wanneer hy geen wins maak nie?
GR 10 MATHEMATICS TUTORIAL

      10.1.2: Exponents, Surds, Rounding.

            Apply laws of exponents in order to simplify expressions with integral indices.
            Find the positional value of a surd in terms of two integers.
            Round off rational and irrational numbers.

      Summary of laws of exponents:

              a m  a n  a mn
              am
                  a mn
              an
                            an
              a 0  a nn       1
                            an
                              a0    1
             a n  a 0n  n  n
                              a     a
              (ab)  a b
                   n      n n

                   n
              a    an
                 n
              b    b
              (a )  a mn
                m n




1.     Use laws of exponents to simplify the expressions below:

1.1      am x an  ap                                   1.2     [(am )n ]p

                                                                                             2 3.2 4
1.3      [(am )n ]p                                     1.4     2-3 + ( a + b )0 x
                                                                                               2 5
           a 2   3
1.5        0
           2b                                         1.6     [(a -2 ) -3 ] –5
                

          2a 3b 2  8a 5 b 10                                  2 x   3
                                                                             y 2   3x 1 y 3 
                                                                                   3                    2


                                                                              6 x         y2 
1.7                                                     1.8
               2a  2 b 7                                                              2       3



                  1
          (a  x) 2 . a 2  x 2
1.9                      1
                                   give your answer with positive exponents.
               (a  x)   2
1.10
         3 p q   3 p
                 2   1 3          1
                                        q2 
                                           2


                 pq 
                                                give your answer with positive exponents.
                            2 4



2.
Try this

Simplify the following compound algebraic fractions as far as possible:
                 6
        5n  7 
1.               n
            3 8
       5 2 
           n     n

             1
2.
                 1
       1
                   1
            1
                 x 1

3.     Consider the small squares with sides x cm, that are put together to form diagrams
       M and N alongside.
3.3    Which diagram has an area of 3x 2 ?                                                  3x
3.4    Which diagram has an area of 3x 2 ?
3.3    Do you think 3x 2  3x 2 ? Explain.                          3x             3x
3.4    What is the area of M and N if x = 1, 2 and 3?
                                                               x
3.5    What value of x will make the areas of M and N
                                                                      M                     N
       the same? Solve the problem:
       a)     Graphically
       b)     Algebraically

4.     Solve for x without the use of a calculator:
4.1    25x+1 = 125
4.2    82x = 32x+1
       8 x 1
4.3              16 1
        4x
4.4    10 ( x  4)( x  2)  0,00001

5.     Suppose that you are the teacher of the new grade 10 Mathematics class. Write a
       summary of the rules and definitions you need for simplifying indices. You need
       to give an example in each case so that any new and inexperienced teacher
       (novice) can present the work in case you need to attend a teacher conference or
       workshop.

6.     Explain the difference between a square root and a surd.
                            7.       al’Khwarizmi, whose full name is Abu Abd-Allah ibn
                             Musa al’Khwarizmi, was born about AD 790 near Baghdad,
                           and died about 850. His most important contribution, written in
                            830, was Hisab al-jabr w’al-muqabala. From the al-jabr in the
                             title we get algebra which is synonymous with the science of
                            equations. His writings led to the discovery of a method and a
                            formula for approximating the square root of a number x. The
                                                    x  p2
                            formula is x  p               where p is the square root of the
                                                    2p 1
                                            nearest perfect square less than x.

Example: Use the formula above to approximate the square root of 19.
Since 16 is the perfect square that is less than 19, then p  16  4 .
           x  p2
     x  p
           2p 1
           19  16
 19  4 
           2(4)  1
            3
       4
            9
       4,333
Note that the answer does not differ much from that of a calculator, which is 4,359

7.1      Now use this formula to estimate:
         a).   27                                           b).    3 67

7.2      Check your answers in 7.1 using your calculator

8.       Given the numbers:     56 ;    132 ;       379 ;   4000
8.1      Estimate where these numbers will be on the number line and justify your
         reasoning without using a calculator.
8.2      Now determine the value of each number correct to two decimal places with the
         use of a calculator.

9.       Without using a calculator, determine where the answer to the following
         expression would fit on the number line.
          1    3     3 1 2 
            of 6 1     3  1 1
                    2
          2    4     2 3 3  2
                                                                   x              183
10.     Mrs. Naicker, the owner of a butchery in Knysna,
        wants to double the area of the existing rectangular
        parking area for her customers. Currently the                            16 836 m 2       92
        dimensions are 183 m by 92 m. For the parking
        area to be doubled, both the length and width
        should be increased by x metres. Determine the
                                                                                                  x
        dimensions of the new parking area.



11. Sipho has to build a triangular tile whose sides                   A
    are all equal.

11.1    Find the value of x if AB AC                                                3x  2
11.2.   Can this triangle be equilateral? Explain
                                                          5x  8

                                                                                              C

                                                               B              x 9

12.     A company has the following production cost function (C) and revenue function
        (R) that they use to produce and sell x copies of a new youth magazine:
        C(x)  10x  600
        R(x)  80x  x2
        Determine how many copies of the magazine must be sold for the production cost
        and revenue to break even.

13.     To determine the profit he makes on a full load of his lorry, a cartage contractor
                                3
        uses the formula P   x 2  12x , where P is the profit (in rands) per hour and
                                20
        x, the speed at which he travels (in km/h).

13.1    What is his speed when his profit is R180?
13.2    Use your graph to answer the following questions:
        a)   What is his maximum profit?
        b)   How fast must he travel to reach his maximum profit?
        c)   What is his speed when he makes no profit?

				
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