### Numbers: Powers and roots

### Fractions raised to an integral power

We have seen what *powers* with an integer as a base look like. Now we will take a fraction as our base.

We can repeatedly multiply a fraction with itself.

\[\begin{array}{rclrc}\left(\frac{\blue2}{\green3}\right)^\orange0&&&=\frac{\blue2^\orange0}{\green3^\orange0}&=1\\\left(\frac{\blue2}{\green3}\right)^\orange1&=&\frac{\blue2}{\green3}&= \frac{\blue2^\orange1}{\green3^\orange1}&=\frac{2}{3}\\\left(\frac{\blue2}{\green3}\right)^\orange2&=&\frac{\blue2}{\green3}\times \frac{\blue2}{\green3}&=\frac{\blue2^\orange2}{\green3^\orange2} &=\frac{4}{9}\\\left(\frac{\blue2}{\green3}\right)^\orange3&=&\frac{\blue2}{\green3} \times \frac{\blue2}{\green3} \times \frac{\blue2}{\green3}&=\frac{\blue2^\orange3}{\green3^\orange3}&=\frac{8}{27} \\ \left(\frac{\blue2}{\green3}\right)^\orange4&=&\frac{\blue2}{\green3} \times \frac{\blue2}{\green3} \times \frac{\blue2}{\green3} \times \frac{\blue2}{\green3}&=\frac{\blue2^\orange4}{\green3^\orange4}&=\frac{16}{81}\end{array}\]

In general we can state:

*The power of a fraction is the power of the #\blue{\textit{numerator}}# divided by the power of the #\green{\textit{denominator}}#.*

**Examples**

\[\begin{array}{rcl}\left(\frac{\blue1}{\green5}\right)^\orange4&=&\frac{\blue1^\orange4}{\green5^\orange4} \\ &=& \frac{1}{625} \\ \\ \left(\frac{\blue3}{\green4}\right)^\orange2&=&\frac{\blue3^\orange2}{\green4^\orange2}\\ &=& \frac{9}{16}\\ \\ \left(-\frac{\blue1}{\green3}\right)^\orange4&=&\frac{\left(\blue{-1}\right)^\orange4}{\green3^\orange4}\\ &=& \frac{1}{81}\\ \\ \end{array}\]

#\begin{array}{rcl}

\left(\dfrac{2}{3}\right)^{2}&=&\dfrac{2^2}{3^2} \\

&&\phantom{xxx}\blue{\text{taken the power of the numerator and the denominator separately}} \\

&=& \dfrac{2 \times 2}{3 \times 3}\\

& &\phantom{xxx}\blue{\text{exponentiation is repeated multiplication}}\\

&=& \dfrac{4}{9} \\

&&\phantom{xxx}\blue{\text{multiplied}}\\

\end{array}#

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