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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

 
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