# What is the power of a quotient property?

Dec 25, 2014

The Power of a Quotient Rule states that the power of a quotient is equal to the quotient obtained when the numerator and denominator are each raised to the indicated power separately, before the division is performed.
i.e.: ${\left(\frac{a}{b}\right)}^{n} = {a}^{n} / {b}^{n}$
For example:
${\left(\frac{3}{2}\right)}^{2} = {3}^{2} / {2}^{2} = \frac{9}{4}$

You can test this rule by using numbers that are easy to manipulate:
Consider: $\frac{4}{2}$ (ok it is equal to $2$ but for the moment let it stay as a fraction), and let us calculate it with our rule first:
${\left(\frac{4}{2}\right)}^{2} = {4}^{2} / {2}^{2} = \frac{16}{4} = 4$
Let us, now, solve the fraction first and then raise to the power of $2$:
${\left(\frac{4}{2}\right)}^{2} = {\left(2\right)}^{2} = 4$

This rule is particularly useful if you have more difficult problems such as an algebraic expression (with letters):
Consider: ${\left(\frac{x + 1}{4 x}\right)}^{2}$
You can now write:
${\left(\frac{x + 1}{4 x}\right)}^{2} = {\left(x + 1\right)}^{2} / {\left(4 x\right)}^{2} = \frac{{x}^{2} + 2 x + 1}{16 {x}^{2}}$