Integer Exponents and Rules of 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

 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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