If the base of the power is a quotient, like in [[$ \left( \dfrac{12}{3} \right)^2 $]], the expression is called the power of a quotient. The value of a power can be calculated using normal calculation rules [[$ \left( \dfrac{12}{3} \right)^2 = (4)^2 = 16 $]], or by first raising both the numerator and the denominator to the second power: [[$ \left( \dfrac{12}{3} \right)^2 = \dfrac{12^2}{3^2} = \dfrac{144}{9} = 16 $]].

The power of a quotient

The power of a quotient is the quotient of the exponents.

[[$ \left( \dfrac{a}{b} \right)^n = \dfrac{a^n}{b^n} $]], [[$ b \neq 0 $]].

The rules for calculating the power of a quotient do not necessarily have to be used when calculating with numbers. However, expressions that include variables cannot be simplified according to normal calculation rules.

Examples

Example 1

Simplify the expressions.

a) [[$ \left( \dfrac{3}{4} \right)^2 = \dfrac{3^2}{4^2} = \dfrac{9}{16} $]]

b) [[$ \left( \dfrac{x}{6} \right)^2 = \dfrac{x^2}{6^2} = \dfrac{x^2}{36} $]]

c) [[$ \left( \dfrac{3x}{5} \right)^2 = \dfrac{3^2 \cdot x^2}{5^2} = \dfrac{9x^2}{25} $]]

d) [[$ \left( \dfrac{2a^3 \cdot a^2}{b \cdot b^5} \right)^4 = \left( \dfrac{2a^{3+2}}{b^{1+5}}\right)^4 = \left( \dfrac{2a^5}{b^6} \right)^4 = \dfrac{2^4 \cdot a^{5\cdot4}}{b^{6\cdot4}} = \dfrac{16a^{20}}{b^{24}} $]]

Example 2

Calculate the expressions by first simplifying them to a single power.

a) [[$ \dfrac{16^3}{8^3} = \left( \dfrac{16}{8} \right)^3 = 2^3 = 8 $]]

b) [[$ \dfrac{15^3}{5^3} = \left( \dfrac{15}{5} \right)^3 = 3^3 = 27 $]]

Example 3

Express as a single power.

a) [[$ 5^7 \cdot \left( \dfrac{1}{6} \right)^7 = \left( 5 \cdot \dfrac{1}{6} \right)^7 = \left( \dfrac{5}{6} \right)^7 $]]

b) [[$ \dfrac{4a^2}{3^2} = \dfrac{2^2a^2}{3^2} = \left( \dfrac{2a}{3} \right)^2 $]]