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.

\displaystyle \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}

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

General Explanation	Example
$\def\arraystretch{1.25} \begin{aligned}\left(\frac ab\right)^n&=\underbrace{\left(\frac ab\right)\cdot\left(\frac ab\right)\cdot\;…\;\cdot\left(\frac ab\right)}_{\text{n times}}\\&=\underbrace{\frac ab\cdot\frac ab\cdot\;…\;\cdot\frac ab}_{\text{n times}}\\&=\frac{\overbrace{a\cdot a\cdot\;…\;\cdot a}^{\text{n times}}}{\underbrace{b\cdot b\cdot\;…\;\cdot b}_{\text{n times}}}\\&=\frac{a^n}{b^n}\end{aligned}$	$\def\arraystretch{1.25} \begin{aligned}\left(\frac57\right)^4&=\underbrace{\left(\frac57\right)\cdot\left(\frac57\right)\cdot\left(\frac57\right)\cdot\left(\frac57\right)}_{\text{4 times}}\\&={\textstyle\underbrace{{\displaystyle\frac57}\cdot{\displaystyle\frac57}\cdot{\displaystyle\frac57}\cdot\displaystyle\frac57}_{\text{4 times}}}\\&=\frac{\overbrace{5\cdot5\cdot5\cdot5}^{\text{4 times}}}{\underbrace{7\cdot7\cdot7\cdot7}_{\text{4 times}}}\\&=\frac{5^4}{7^4}\end{aligned}$

General Explanation

Example

$\def\arraystretch{1.25} \begin{aligned}\left(\frac ab\right)^n&=\underbrace{\left(\frac ab\right)\cdot\left(\frac ab\right)\cdot\;…\;\cdot\left(\frac ab\right)}_{\text{n times}}\\&=\underbrace{\frac ab\cdot\frac ab\cdot\;…\;\cdot\frac ab}_{\text{n times}}\\&=\frac{\overbrace{a\cdot a\cdot\;…\;\cdot a}^{\text{n times}}}{\underbrace{b\cdot b\cdot\;…\;\cdot b}_{\text{n times}}}\\&=\frac{a^n}{b^n}\end{aligned}$

$\def\arraystretch{1.25} \begin{aligned}\left(\frac57\right)^4&=\underbrace{\left(\frac57\right)\cdot\left(\frac57\right)\cdot\left(\frac57\right)\cdot\left(\frac57\right)}_{\text{4 times}}\\&={\textstyle\underbrace{{\displaystyle\frac57}\cdot{\displaystyle\frac57}\cdot{\displaystyle\frac57}\cdot\displaystyle\frac57}_{\text{4 times}}}\\&=\frac{\overbrace{5\cdot5\cdot5\cdot5}^{\text{4 times}}}{\underbrace{7\cdot7\cdot7\cdot7}_{\text{4 times}}}\\&=\frac{5^4}{7^4}\end{aligned}$

Further examples

$\left(\frac34\right)^2=\frac{3\cdot3}{4\cdot4}=\frac{3^2}{4^2}=\frac9{16}$
$\left(\frac23\right)^3=\frac{2^3}{3^3}=\frac8{27}$

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.

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

More explicitly: $\left(-\frac34\right)^2=\frac{\left(-3\right)^2}{\left(4\right)^2}=\frac9{16}$

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

More explicitly: $\left(-\frac34\right)^3=\frac{\left(-3\right)^3}{\left(4\right)^3}=\frac{-27}{64}=-\frac{27}{64}$

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