Express each of the following rational numbers with a positive exponent:

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Question:

Express each of the following rational numbers with a positive exponent:

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

(ii) $\left(\frac{5}{4}\right)^{-3}$

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

(iv) $\left\{\left(\frac{4}{3}\right)^{-3}\right\}^{-4}$

(v) $\left\{\left(\frac{3}{2}\right)^{4}\right\}^{-2}$

Solution:

$(i)\left(\frac{3}{4}\right)^{-2} \quad \cdots\left(a^{-1}=1 / a\right)$

$=\left(\frac{4}{3}\right)^{2}$

$(i i)\left(\frac{5}{4}\right)^{-3} \quad \cdots\left(a^{-1}=1 / a\right)$

$=\left(\frac{4}{5}\right)^{3}$

$(i i i) 4^{3} \times 4^{-9} \quad \cdots\left(a^{m} \times a^{n}=a^{m+n}\right)$

$=4^{(3-9)}=4^{-6}$

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

$(i v)\left\{\left(\frac{4}{3}\right)^{-3}\right\}^{-4} \quad \cdots\left(\left(a^{m}\right)^{n}=a^{m n}\right)$

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

$=\left(\frac{4}{3}\right)^{12}$

$(v)\left\{\left(\frac{3}{2}\right)^{4}\right\}^{-2} \cdots\left(\left(a^{m}\right)^{n}=a^{m n}\right)$

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

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

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

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