Integer Exponents and Rules of Exponents

Integer Exponents


Definition:  

                   1
       x
-1 =       
                   x

and 

                    1
       x
-n =         
                   xn

 

Example

       
3 -1  =  1/3

         2
-3  =  1/23  =  1/8


Definition:  

     x0  =  1 

for all nonzero x.

 


Exponent  Rule 1

 

Addition Rule

       xn xm  =  xn+ m 

 

Examples:  

  1. 23 2  =   23 + 5   =  28

  2. w2 w3  =  w5

  3. xy2 x3y3x4y4  =  x8y9



Rule 2


The Power Rule

       (xn)m  =   xnm

 

Examples:

  1. (24)2  =  28

  2. (x2)3  =  x6 

  3. 42 23 8  =  (22)2 23 23  =  24 23 23  =  210  


Rule III


The Distributive Rule

       (xy)n  =  xnyn    

 

Examples:

  1. (6)4 = (2)4(3)4

  2. (3x2)3 = 33 (x2)3  = 27 x6    

  3. (-x3)4  =  (-1)4 (x3)4  =  (1) (x12)  =  x12


Rule IV


The Quotient to Difference Rule

For x not equal to 0,

        xn
                 =  xn-m  
       xm

 

Examples

  1.    x3
               =  x3-2  =  x1  =  x
       x2 

  2.    x4                                   1
              =  x4 - 8  =  x-4  =           
       x8                                  x4                          

 

Application

The population in billions of the world can be modeled by the equation

        P  =  A 2t/30

where t is the time in years after 1970.  If there were 3 billion people in 1970, What will the population be in 2060?

 

Solution:  

We first know that if t is 0 then P is 3 (in billions).  Hence

        3  =  A 20/30  =  A

So that

        P  =  3 (2t/30)

2060 corresponds to t = 90 so that

        P  =  (3) (290/30)   =  (3) (23)  =  (3)(8)  =  24 billion people.

 



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