Definition and properties
Let $n\in\mathbb{Z}$. A power of a fraction is the fraction multiplied by itself $n$ times:
$$\left(\dfrac{a}{b}\right)^{n}=\underbrace{\dfrac{a}{b}\cdot\dfrac{a}{b}\cdots\dfrac{a}{b}}_{n\ \text{times}}=\dfrac{a^{n}}{b^{n}}$$
| # | Property |
|---|---|
| (1) | $\left(\dfrac{a}{b}\right)^{0}=1$ |
| (2) | $\left(\dfrac{a}{b}\right)^{1}=\dfrac{a}{b}$ |
| (3) | $\left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}$ |
| (4) | $\left(\dfrac{a}{b}\right)^{n}\cdot\left(\dfrac{a}{b}\right)^{m}=\left(\dfrac{a}{b}\right)^{n+m}$ |
| (5) | $\left(\dfrac{a}{b}\right)^{n}:\left(\dfrac{a}{b}\right)^{m}=\left(\dfrac{a}{b}\right)^{n-m}$ |
| (6) | $\left[\left(\dfrac{a}{b}\right)^{n}\right]^{m}=\left(\dfrac{a}{b}\right)^{n\cdot m}$ |
| (7) | $\left(\dfrac{a}{b}\right)^{n}\cdot\left(\dfrac{c}{d}\right)^{n}=\left(\dfrac{a}{b}\cdot\dfrac{c}{d}\right)^{n}$ |
| (8) | $\left(\dfrac{a}{b}\right)^{n}:\left(\dfrac{c}{d}\right)^{n}=\left(\dfrac{a}{b}:\dfrac{c}{d}\right)^{n}$ |
| (9) | $\left(-\dfrac{a}{b}\right)^{n}=\begin{cases}-\left(\dfrac{a}{b}\right)^{n} & n\ odd\\[4pt] \left(\dfrac{a}{b}\right)^{n} & n\ even\end{cases}$ |
Examples
The left column shows the property used at each step.
| $\left[\left(\dfrac{3}{2}\right)^{2}\right]^{3}:\left(\dfrac{4}{6}\right)^{-1}\cdot\left(\dfrac{9}{4}\right)^{-3}$ | |
| (6) | $\left(\dfrac{3}{2}\right)^{6}:\left(\dfrac{4}{6}\right)^{-1}\cdot\left(\dfrac{9}{4}\right)^{-3}$ |
| (3) | $\left(\dfrac{3}{2}\right)^{6}:\left(\dfrac{6}{4}\right)^{1}\cdot\left(\dfrac{9}{4}\right)^{-3}$ |
| simplify | $\left(\dfrac{3}{2}\right)^{6}:\left(\dfrac{3}{2}\right)^{1}\cdot\left(\dfrac{9}{4}\right)^{-3}$ |
| (5) | $\left(\dfrac{3}{2}\right)^{5}\cdot\left(\dfrac{9}{4}\right)^{-3}$ |
| $\tfrac{9}{4}=\left(\tfrac{3}{2}\right)^2$ and (6) | $\left(\dfrac{3}{2}\right)^{5}\cdot\left[\left(\dfrac{3}{2}\right)^{2}\right]^{-3}=\left(\dfrac{3}{2}\right)^{5}\cdot\left(\dfrac{3}{2}\right)^{-6}$ |
| (4) | $\left(\dfrac{3}{2}\right)^{-1}$ |
| (3) | $\dfrac{2}{3}$ |
With a negative base: the exponent $3$ is odd, so the minus sign comes out with property (9).
| $\left(-\dfrac{2}{5}\right)^{3}\cdot\left(\dfrac{25}{4}\right)^{-4}:\left(\dfrac{1}{3}\right)^{11}$ | |
| (9), (3) | $-\left(\dfrac{2}{5}\right)^{3}\cdot\left(\dfrac{4}{25}\right)^{4}:\left(\dfrac{1}{3}\right)^{11}$ |
| $\tfrac{4}{25}=\left(\tfrac{2}{5}\right)^2$ | $-\left(\dfrac{2}{5}\right)^{3}\cdot\left[\left(\dfrac{2}{5}\right)^{2}\right]^{4}:\left(\dfrac{1}{3}\right)^{11}$ |
| (6) | $-\left(\dfrac{2}{5}\right)^{3}\cdot\left(\dfrac{2}{5}\right)^{8}:\left(\dfrac{1}{3}\right)^{11}$ |
| (4) | $-\left(\dfrac{2}{5}\right)^{11}:\left(\dfrac{1}{3}\right)^{11}$ |
| (8) | $-\left(\dfrac{2}{5}:\dfrac{1}{3}\right)^{11}$ |
| $-\left(\dfrac{6}{5}\right)^{11}$ |