EDGE Math 8

Zero and Negative Integer Exponents

Negative Exponents
To define a negative integer exponent, consider $a^{- n}$ , where $a$ is a nonzero real number and $n$ is a positive integer. If the Product Rule must hold for $m = - n$ , then

a^{- n} \cdot a^{n} = a^{- n + n} = a^{0} = 1.

Hence, $a^{- n} \cdot a^{n} = 1.$ For this equation to be true, $a^{- n}$ and $a^{n}$ should be reciprocals of each other. Thus, $a^{- n} = \frac{1}{a^{n}} .$

Definition

If $a$ is a nonzero real number and $n$ is a positive integer, then

a^{- n} = \frac{1}{a^{n}} .

Example 1

Simplify each expression.

a. $5^{- 2}$

b. ${(- 2)}^{- 4}$

c. $y^{- 6}$ , where $y \neq 0$

d. ${(- b)}^{- 3}$ , where $b \neq 0$

Solution

Use the definition of negative integer exponent.

a. $5^{- 2} = \frac{1}{5^{2}} = \frac{1}{25}$

b. ${(- 2)}^{- 4} = \frac{1}{{(- 2)}^{4}} = \frac{1}{16}$

c. $y^{- 6} = \frac{1}{y^{6}}$

d. ${(- b)}^{- 3} = \frac{1}{{(- b)}^{3}} = \frac{1}{- b^{3}} = - \frac{1}{b^{3}}$