INTEGER EXPONENTS

Let us consider a power with positive integer exponent, say 5³. To simplify 5³, use 5 as a factor two times :

5³  =  5 ⋅ 5 ⋅ 5  =  125

If the exponent is negative or zero, what does it mean ?

A table can be used to look for a pattern to figure it out. 

When the exponent decreases by one, the value of the power is divided by 5. Continue the pattern of dividing by 5.   

5⁰  =  5/5  =  1

5^-1  =  1/5  =  1/5¹

5^-2  =  1/5 ÷ 5  =  1/5²

Zero Exponent

Words :

Any nonzero number raised to the zero power is 1. 

Numbers :

5⁰  =  1

(-5)⁰  =  1

215⁰  =  1

(2/3)⁰  =  1 

Algebra :

If x ≠ 0, then x⁰ = 1. 

Negative Exponent

Words :

Any nonzero number raised to a negative exponent is equal to 1 divided by that number raised to the opposite (positive) exponent.   

Numbers :

5^-2  =  1/5²  =  1/25

2^-4  =  1/2⁴  =  1/16 

Algebra :

If x ≠ 0 and n is an integer, then x^-n = 1/xⁿ. 

Manufacturing Application

The diameter for a piston could vary by at most 10^-3 inch. 

Simplify this expression. 

10^-3  =  1/10³  =  1/(10 ⋅ 10 ⋅ 10)  =  1/1000

10^-3 inch is equal to 1/1000 inch, or 0.0001 inch.    

Zero and Negative Exponents

Example 1 :

Simplify : 

2^-3

Solution :

=  2^-3

=  1 / 2³

=  1 / (2 ⋅ 2 ⋅ 2)

=  1/8

Example 2 :

Simplify : 

7⁰

Solution :

=  7⁰

Any non zero number raised to the zero power is 1. 

=  1

Example 3 :

Simplify : 

(-3)^-4

Solution :

=  (-3)^-4

=  1 / (-3)⁴

=  1 / [(-3)(-3)(-3)(-3)]

=  1/81

Example 4 :

Simplify : 

-3^-4

Solution :

=  -3^-4

=  -1 / 3⁴

=  -1 / (3 ⋅ 3 ⋅ 3 ⋅ 3)

=  -1/81

Evaluating Expressions with Zero and Negative Exponents

Example 5 :

Evaluate x^-1 for x = 3. 

Solution :

=  x^-1

Substitute 3 for x. 

=  3^-1

=  1/3

Example 6 :

Evaluate x⁰y^-3 for x = 8 and y = -2. 

Solution :

=  x⁰y^-3

Substitute 0 for x and -2 for y. 

=  8⁰ ⋅ (-2)^-3

=  1 ⋅ 1/(-2)³

=  1 ⋅ 1 / [(-2)(-2)(-2)]

=  1 ⋅ 1 / (-8)

=  1 ⋅ (-1/8)

=  -1/8

Simplifying Expressions with Zero and Negative Exponents

Example 7 :

Simplify :

5x^-2

Solution :

=  5x^-2

=  5/x²

Example 8 :

Simplify :

-4/k^-4

Solution :

=  -4/k^-4

=  -4 ⋅ 1/k^-4

=  -4 ⋅ k⁴

=  -4k⁴

Example 9 :

Simplify :

a^-3/x⁰b⁵

Solution :

=  a^-3/x⁰b⁵

=  1 / (a³ ⋅ 1 ⋅ b⁵)

=  1/a³b⁵

Example 10 :

Simplify :

2r⁰m^-5

Solution :

=  2r⁰m^-5

=  2 ⋅ 1 ⋅ 1/m⁵

=  2/m⁵

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