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Express each of the following as a rational number of the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q≠0$:
(i) \( 2^{-3} \)
(ii) \( (-4)^{-2} \)
(iii) \( \frac{1}{3^{-2}} \)
(iv) \( \left(\frac{1}{2}\right)^{-5} \)
(v) \( \left(\frac{2}{3}\right)^{-2} \)
To do:
We have to express each of the given rational numbers as a rational number of the form $\frac{p}{q}$.
Solution:
We know that,
$a^{-m}=\frac{1}{a^m}$
$a^{m}=\frac{1}{a^{-m}}$
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)$\frac{1}{2^{-5}}=2^5$
$=32$ 
(v) $(\frac{2}{3})^{-2}=\frac{1}{(\frac{2}{3})^{2}}$
$=(\frac{3}{2})^{2}$  
$=\frac{3^2}{2^2}$
$=\frac{9}{4}$
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