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In this section we will extend the definition of exponents to include all integers and to learn some rules for working with integral exponents.

Positive and Negative Exponents

Positive integral exponents provide a convenient way to write repeated multiplication or very large numbers. For example,

2 · 2 · 2 = 23, y · y · y · y = y4, and 1,000,000,000 = 109.

We refer to 23 as “2 cubed,” “2 raised to the third power,” or “a power of 2.”

Positive Integral Exponents

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

In the exponential expression an, the base is a, and the exponent is n.

We use 2-3 to represent the reciprocal of 23. Because 23 = 8, we have . In general, a-n is defined as the reciprocal of an.

Negative Integral Exponents

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

(If n is positive, -n is negative.)

To evaluate 2-3, you can first cube 2 to get 8 and then find the reciprocal to get , or you can first find the reciprocal of 2 (which is ) and then cube to get . So

The power and the reciprocal can be found in either order. If the exponent is -1, we simply find the reciprocal. For example,

Because 23 and 2-3 are reciprocals of each other, we have

These examples illustrate the following rules.

Rules for Negative Exponents

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

Example 1

Negative exponents

Evaluate each expression.

a) 3-2

b) (-3)-2

c) -3-2

d)

e)

Solution

Definition of negative exponent
Definition of negative exponent
Evaluate 3-2, then take the opposite.
The reciprocal of .

The cube of .

The reciprocal of 5-3 is 53.

Caution

We evaluate -32 by squaring 3 first and then taking the opposite. So -32 = -9, whereas (-3)2 = 9. The same agreement also holds for negative exponents. That iswhy the answer to Example 1(c) is negative.