Question:
Express each of the following as a rational number of the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
(i) $2^{-3}$
(ii) $(-4)^{-2}$
(iii) $\frac{1}{3^{-2}}$
(iv) $\left(\frac{1}{2}\right)^{-5}$
(v) $\left(\frac{2}{3}\right)^{-2}$
Solution:
We know that $\mathrm{a}^{-\mathrm{n}}=\frac{1}{\mathrm{a}^{\mathrm{n}}}$. Therefore,
(i) $2^{-3}=\frac{1}{2^{3}}=\frac{1}{8}$
(ii) $(-4)^{-2}=\frac{1}{(-4)^{2}}=\frac{1}{16}$
(iii) $\frac{1}{3^{-2}}=3^{2}=9$
(iv) $\left(\frac{1}{2}\right)^{-5}=2^{5}=32$
(v) $\left(\frac{2}{3}\right)^{-2}=\left(\frac{3}{2}\right)^{2}=\frac{9}{4}$
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