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# Raising a Quotient to a Power

Consider an example of applying known rules to a power of a quotient:

We get a similar result with a negative power:

In each of these cases the original exponent applies to both the numerator and denominator. These examples illustrate the power of a quotient rule.

Power of a Quotient Rule

If a and b are nonzero real numbers and n is any integer, then

Example 1

Using the power of a quotient rule

Use the rules of exponents to simplify each expression. Write your answers with positive exponents only. Assume the variables are nonzero real numbers.

Solution

 Power of a quotient rule
 Because (x3)3 = x9 and (y2)3 = y6

A fraction to a negative power can be simplified by using the power of a quotient rule as in Example 3. Another method is to find the reciprocal of the fraction first, then use the power of a quotient rule as shown in the next example.

Example 2

Negative powers of fractions

Simplify. Assume the variables are nonzero real numbers and write the answers with positive exponents only.

Solution

 The reciprocal of Power of a quotient rule