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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Updated on: 10-Oct-2022

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