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

Examples:  

1. 2³ 2⁵   =   2^{3 + 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
              =  x^3-2  =  x¹  =  x
      x² 
   

2.    x⁴                                   1
             =  x^{4 - 8}  =  x^-4  =           
      x⁸                                  x⁴                          

Application

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

        P  =  A 2^t/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 2^0/30  =  A

So that

        P  =  3 (2^t/30)

2060 corresponds to t = 90 so that

        P  =  (3) (2^90/30)   =  (3) (2³)  =  (3)(8)  =  24 billion people.

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