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					Laws of Exponents

 x *x  x
  4   8   12


                 ( a * b)  a * b
                        3      3     3


          3113
                315
          318



                       By Chancy Nordick
Review
What exactly is a power function again?

 A power function has two parts:


        y
    x
Review Continued
Where
    x  x1 * x2 ........* xn 1 * xn
        n




                     And



                b     1
            x         b
                      x
Multiplication Laws
What if we want to multiply two power functions?

                    2 4
     Example:     x x

             ( xx)( xxxx)

              xxxxxx
                       6
                   x
Multiplication Laws
In General

                  a b
       x x x
         a   b



  Note: Bases must be the same!
                                                  a b
A few examples                  x x x
                                     a   b




                 4 7
   2 *2  2
       4   7
                        2      11



                     3 x 4 x
  3x
 b *b      4x
                b                      b   7x
First Law
 xm*xn = xm+n
 Notice that the bases have to be the same.
 Find    x2*x3
 Find    x3y*x4y7
 Find   z2x5y9*zx7y4
Multiplication Laws
Now, what about powers of powers?
  Example:         2 3
                 (a )

             2       2   2
         a *a *a

      (aa )(aa )(aa )
                     6
                 a
Multiplication Laws
In General



       (x )  x
             a b      ab
A few examples            (x )  x
                            a b      ab




   2 2
         6
     3         3*6
                     2    18


   x  x
    28       2*8
                   x16
The Third law
 (xm)n = xmn
 Find   (x4)5
 Find   (x3y9)8
 Find   (2x3y4)3
Multiplication Laws
Finally, how do I deal with powers of a product?
                           3
   Example     (h * j )

         (h * j )(h * j )(h * j )

         h*h*h* j * j * j

                3      3
              h *j
Multiplication Laws
In General



     ( xy )  x y
             a        a   a
Second Law
 (xy)m = xmym
 The base does not have to be the same here.
 Find   (xy)7
 Find   (xy)5
 Find   (2x)3
Quick Review
               a b
     x x x
       a   b


    (x )  x
      a b    ab


   ( xy )  x y
           a   a      a
A few examples
  (3* x )  9x 2
             2




  (ax)  (ax )  a x
        23           6    6       6



  (2t )3 ( t 2 )  8(t 3 )(t 2 )  8t 5


  4 *2  2 * 2  2 *2  23 x
    x            x   2x       x       2x   x
We’re done….



 With the multiplication laws, before
 We go on….
Stand up and Stretch!
Onward!



 to the division laws
Division Laws
Example



   7
 x     xxxxxxx
   3
               xxxx  x 4

 x       xxx
Division Laws
In General

             a
        x      a b
          b
             x ,x  0
        x
                                a
A few Examples              x      a b
                              b
                                 x ,x  0
                            x
    8
   l           8 6
   l 6
          l          l    2


    x 4
   3           ( x 4) 4
    34
          3                3      x
The Fourth Law
 xm/xn =      xm-n
 The base has to be the same.
 Find     x5/x3
 Find    x3y2/x2y
Division Laws
Similar to Multiplication


      3
  h   h h h h3
   j  * *  3
      j j j  j
Division Laws
In General

             a
        x   x  a

         y  a ,y0
            y
                         a
                     x     xa
Examples              y   ya , y  0
                      


                 3
      4
           3
               4
              3
      5      5

           4     4
     x       x
       
     7       7 4
Quick Review of Division Laws
      a
     x      a b
       b
          x ,x  0
     x
          a
     x     x a

      y   ya , y  0
      
 Laws of Exponents

 Compute:        2 5



Answer: 32
 Laws of Exponents

 Compute:        5 -2



Answer: 1/25
 Laws of Exponents

 Compute:        0 2



Answer: 0
 Laws of Exponents

 Compute:        2 0



Answer: 1
 Laws of Exponents

 Compute:        0 0



Answer: ??
 Laws of Exponents

 Compute: -          42



Answer: -16
 Laws of Exponents

 Compute:        (-4) 2



Answer: 16
 Laws of Exponents

 Compute: -          4-2



Answer: - 1/16
 Laws of Exponents

 Compute:        0.125 -2



Answer: 64
 Laws of Exponents

 Compute:          (4 5 )(42)



Answer:    47 or   16,384
Laws of Exponents

Compute:        (4 5 )2



    Answer:    4 10 or

     10,485,676
 Laws of Exponents


What is one-third of
        399 ?

      Answer:    3 98

				
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