Negative Integral Exponents

If x is nonzero, the reciprocal of x is written as For example, the reciprocal of 2³ 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 · 3² = 1, the reciprocal of 3 ^-2 is 3², 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