Question

Question 6(ii)
Evaluate:
${\left(\frac{5}{8}\right)}^{-7}\times {\left(\frac{8}{5}\right)}^{-4}$

Answer 1

${\left(\frac{5}{8}\right)}^{-7}\times {\left(\frac{8}{5}\right)}^{-4}=\frac{5^{-7}}{8^{-7}}\times \frac{8^{-4}}{5^{-4}}[\because {\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}]=5^{-7-(-4)}\times 8^{-4-(-7)}[\because a^m÷a^n=a^{m-n}]=5^{-7+4}\times 8^{-4+7}=5^{-3}\times 8^3=\frac{8^3}{5^3}[\because a^{-m}=\frac{1}{a^m}]=\frac{512}{125}$

Alternative:

${\left(\frac{5}{8}\right)}^{-7}\times {\left(\frac{8}{5}\right)}^{-4}$
= ${\left(\frac{8}{5}\right)}^7\times {\left(\frac{8}{5}\right)}^{-4}[\because a^{-m}=\frac{1}{a^m}]$
= ${\left(\frac{8}{5}\right)}^{7+(-4)}[\because a^m\times a^n=a^{m+n}]$
= ${\left(\frac{8}{5}\right)}^{(7-4)}$
= ${\left(\frac{8}{5}\right)}^3$
=$\frac{512}{125}$