Exponentiating Fractions

In this article you will learn how to exponentiate fractions - both theoretically and with examples.

You take the n-th power of a fraction by taking the n-th power of the denominator and numerator separately.

{(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}

{(\frac{5}{7})}^{4} = \frac{5^{4}}{7^{4}} = \frac{625}{2401}

General Explanation	Example
$\begin{aligned} {(\frac{a}{b})}^{n} & = \underset{n times}{\underset{⏟}{(\frac{a}{b}) \cdot (\frac{a}{b}) \cdot \dots \cdot (\frac{a}{b})}} \\ = \underset{n times}{\underset{⏟}{\frac{a}{b} \cdot \frac{a}{b} \cdot \dots \cdot \frac{a}{b}}} \\ = \frac{\overset{n times}{\overset{⏞}{a \cdot a \cdot \dots \cdot a}}}{\underset{n times}{\underset{⏟}{b \cdot b \cdot \dots \cdot b}}} \\ = \frac{a^{n}}{b^{n}} \end{aligned}$	$\begin{aligned} {(\frac{5}{7})}^{4} & = \underset{4 times}{\underset{⏟}{(\frac{5}{7}) \cdot (\frac{5}{7}) \cdot (\frac{5}{7}) \cdot (\frac{5}{7})}} \\ = \underset{4 times}{\underset{⏟}{\frac{5}{7} \cdot \frac{5}{7} \cdot \frac{5}{7} \cdot \frac{5}{7}}} \\ = \frac{\overset{4 times}{\overset{⏞}{5 \cdot 5 \cdot 5 \cdot 5}}}{\underset{4 times}{\underset{⏟}{7 \cdot 7 \cdot 7 \cdot 7}}} \\ = \frac{5^{4}}{7^{4}} \end{aligned}$

General Explanation

Example

$\begin{aligned} {(\frac{a}{b})}^{n} & = \underset{n times}{\underset{⏟}{(\frac{a}{b}) \cdot (\frac{a}{b}) \cdot \dots \cdot (\frac{a}{b})}} \\ = \underset{n times}{\underset{⏟}{\frac{a}{b} \cdot \frac{a}{b} \cdot \dots \cdot \frac{a}{b}}} \\ = \frac{\overset{n times}{\overset{⏞}{a \cdot a \cdot \dots \cdot a}}}{\underset{n times}{\underset{⏟}{b \cdot b \cdot \dots \cdot b}}} \\ = \frac{a^{n}}{b^{n}} \end{aligned}$

$\begin{aligned} {(\frac{5}{7})}^{4} & = \underset{4 times}{\underset{⏟}{(\frac{5}{7}) \cdot (\frac{5}{7}) \cdot (\frac{5}{7}) \cdot (\frac{5}{7})}} \\ = \underset{4 times}{\underset{⏟}{\frac{5}{7} \cdot \frac{5}{7} \cdot \frac{5}{7} \cdot \frac{5}{7}}} \\ = \frac{\overset{4 times}{\overset{⏞}{5 \cdot 5 \cdot 5 \cdot 5}}}{\underset{4 times}{\underset{⏟}{7 \cdot 7 \cdot 7 \cdot 7}}} \\ = \frac{5^{4}}{7^{4}} \end{aligned}$

Further examples

${(\frac{3}{4})}^{2} = \frac{3 \cdot 3}{4 \cdot 4} = \frac{3^{2}}{4^{2}} = \frac{9}{16}$
${(\frac{2}{3})}^{3} = \frac{2^{3}}{3^{3}} = \frac{8}{27}$

Negative fractions

If the exponent is an odd number, the fraction remains negative.

If the exponent is an even number, the exponentiated fraction becomes positive.

${(- \frac{3}{4})}^{2} = \frac{9}{16}$

More explicitly: ${(- \frac{3}{4})}^{2} = \frac{{(- 3)}^{2}}{{(4)}^{2}} = \frac{9}{16}$

${(- \frac{3}{4})}^{3} = - \frac{27}{64}$

More explicitly: ${(- \frac{3}{4})}^{3} = \frac{{(- 3)}^{3}}{{(4)}^{3}} = \frac{- 27}{64} = - \frac{27}{64}$

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