ZERO AND NEGATIVE EXPONENTS

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. 

Solved Examples

Example 1 :

Simplify : 

5^-3

Solution :

=  5^-3

=  1 / 5³

=  1 / (5 ⋅ 5 ⋅ 5)

=  1/125

Example 2 :

Simplify : 

8⁰

Solution :

=  8⁰

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

=  1

Example 3 :

Simplify : 

(-2)^-4

Solution :

=  (-2)^-4

=  1 / (-2)⁴

=  1 / [(-2)(-2)(-2)(-2)]

=  1/16

Example 4 :

Simplify : 

-3^-4

Solution :

=  -3^-4

=  -1 / 3⁴

=  -1 / (3 ⋅ 3 ⋅ 3 ⋅ 3)

=  -1/81

Example 5 :

Evaluate y^-1 for y = 2. 

Solution :

=  y^-1

Substitute 2 for y. 

=  2^-1

=  1/2

Example 6 :

Evaluate a⁰b^-3 for a = 8 and b = -2. 

Solution :

=  a⁰b^-3

Substitute 8 for a and -2 for b. 

=  8⁰ ⋅ (-2)^-3

=  1 ⋅ 1/(-2)³

=  1 ⋅ 1 / [(-2)(-2)(-2)]

=  1 ⋅ 1 / (-8)

=  1 ⋅ (-1/8)

=  -1/8

Example 7 :

Simplify :

5a^-2

Solution :

=  5a^-2

=  5 ⋅ 1/a²

=  5/a²

Example 8 :

Simplify :

-5/p^-4

Solution :

=  -5/p^-4

=  -5 ⋅ 1/p^-4

=  -5 ⋅ p⁴

=  -5p⁴

Example 9 :

Simplify :

x^-2/a⁰y³

Solution :

=  x^-2/a⁰y³

=  1 / (x² ⋅ 1 ⋅ y³)

=  1/x²y³

Example 10 :

Simplify :

3p⁰n^-5

Solution :

=  3p⁰n^-5

=  3 ⋅ 1 ⋅ 1/n⁵

=  3/n⁵

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