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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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