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


Because (x^{3})^{3} = x^{9} and (y^{2})^{3}
= y^{6} 



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 



