# Integral Exponents

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 Â· 2 Â· 2 = 2^{3}, y Â· y Â· y Â· y = y^{4}, and 1,000,000,000 = 10^{9}.

We refer to 2^{3} as â€œ2 cubed,â€ â€œ2 raised to the third power,â€ or â€œa power of 2.â€

**Positive Integral Exponents **

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

In the **exponential expression** a^{n}, the **base** is a, and the
**exponent** is n.

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

**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 2^{3} 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 **

**Caution**

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