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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. 