Express each of the following rational numbers with a positive exponent:
(i) $ \left(\frac{3}{4}\right)^{-2} $
(ii) $ \left(\frac{5}{4}\right)^{-8} $
(iii) $ 4^{3} \times 4^{-9} $
(iv) $ \left\{\left(\frac{4}{3}\right)^{-3}\right\}^{-4} $
(v) $ \left\{\left(\frac{3}{2}\right)^{4}\right\}^{-2} $

To do:

We have to express the given rational numbers with positive exponents.

Solution:

We know that,

$\frac{1}{a^m}=a^{-m}$ 

$a^{m}=(\frac{1}{a})^{-m}$ 

$(\frac{a}{b})^m=(\frac{b}{a})^{-m}$

$a^m \times a^n=a^{m+n}$

Therefore,

(i) $(\frac{3}{4})^{-2}=(\frac{4}{3})^{2}$

(ii) $(\frac{5}{4})^{-8}=(\frac{4}{5})^{8}$ 

(iii) $4^{3} \times 4^{-9}=4^{3+(-9)}$

$=4^{-6}$ 

$=(\frac{1}{4})^6$

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

$=(\frac{4}{3})^{12}$  

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

$=(\frac{3}{2})^{-8}$

$=(\frac{2}{3})^{8}$ 

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Updated on: 10-Oct-2022

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