Express each of the following as a rational number in the form $\frac{p}{q}$:
(i) $ 6^{-1} $
(ii) $ (-7)^{-1} $
(iii) $ \left(\frac{1}{4}\right)^{-1} $
(iv) $ (-4)^{-1} \times\left(\frac{-3}{2}\right)^{-1} $
(v) $ \left(\frac{3}{5}\right)^{-1} \times\left(\frac{5}{2}\right)^{-1} $

To do:

We have to express the given rational numbers as rational numbers in the form $\frac{p}{q}$.

Solution:

We know that,

$a^{-m}=\frac{1}{a^m}$ 

Therefore,

(i) $(6)^{-1}=\frac{1}{6^1}$

$=\frac{1}{6}$

(ii) $(-7)^{-1}=\frac{1}{(-7)^1}$

$=\frac{-1}{7}$ 

(iii) $(\frac{1}{4})^{-1}=(\frac{4}{1})^1$

$=4$  

(iv) $(-4)^{-1} \times(\frac{-3}{2})^{-1}=(\frac{1}{-4})^{1} \times(\frac{-2}{3})^{1}$

$=\frac{-1}{4} \times \frac{-2}{3}$

$=\frac{-1 \times (-2)}{4 \times 3}$

$=\frac{1}{6}$

(v) $(\frac{3}{5})^{-1} \times(\frac{5}{2})^{-1}=(\frac{5}{3})^{1} \times(\frac{2}{5})^{1}$

$=\frac{5}{3} \times \frac{2}{5}$

$=\frac{2}{3}$

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Updated on: 10-Oct-2022

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