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{1}{4}\right)^{3}&=&\dfrac{1^3}{4^3} \\
&&\phantom{xxx}\blue{\text{taken the power of the numerator and the denominator separately}} \\
&=& \dfrac{1 \times 1 \times 1}{4 \times 4 \times 4}\\
& &\phantom{xxx}\blue{\text{exponentiation is repeated multiplication}}\\
&=& \dfrac{1}{64} \\
&&\phantom{xxx}\blue{\text{multiplied}}\\
\end{array}#
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