# Negative Integral Exponents

If x is nonzero, the reciprocal of x is written as
For example, the reciprocal of 2^{3} is written
To write the reciprocal of an exponential expression in a simpler way, we use a
negative exponent. So
In general we have the following definition.

**Negative Integral Exponents**

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

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

**Example 1**

**Simplifying expressions with negative exponents**

Simplify.

a) 2^{-5}

b) (-2)^{-5}

c)

**Solution**

a)

b)
Definition of negative exponent

c)

**Caution **

In simplifying -5^{-2}, the negative sign preceding the 5 is used
after 5 is squared and the reciprocal is found. So

To evaluate a^{ -n}, you can first find the nth power of a and then
the reciprocal. However, the result is the same if you first find the reciprocal
of a and then find the nth power of the reciprocal. For example,

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 3^{ -2} Â· 3^{2} = 1, the reciprocal of 3^{ -2}
is 3^{2}, and we have

These expamples illustrate the following rules.

**Rules for Negative Exponents**

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

**Example 2**

**Using the rules for negative exponents**

Simplify

**Solution**