Negative Integer Exponents

Definition of a Negative Integer Exponent

You have seen how to work with exponents that are positive integers or 0.

Now, we will investigate exponents that are negative integers.

We’ll begin with this expression:
We can simplify the expression by canceling common factors.
We can also simplify the expression by subtracting exponents.

Since simplifies to both and 2^-3, we conclude .

This relationship holds in general.

Note:

Recall:

2³ = 2 · 2 · 2

2⁰ = 1

Here, x ≠ 0 and n is a nonnegative integer.

Example 1

Find: 5^-2

Solution

Use the definition of a negative exponent:

So,

We can also define as follows:

Definition —

Here, x ≠ 0 and n is a nonnegative integer.

Example 2

a) Find:

b) Find:

Solution

a. Use the definition
b. Use the definition

Note:

A negative exponent does not determine if an expression is positive or negative. For example:

	positive
	negative