Question

By what number should ${(-\frac{2}{3})}^3$ be divided so that the quotient may be equal to ${\left(\frac{9}{4}\right)}^{-2}$?

Answer 1

Let $x$ be the number divided to ${\left(\frac{-2}{3}\right)}^3={\left(\frac{9}{4}\right)}^{-2}$

$\Rightarrow {\left(\frac{-2}{3}\right)}^3÷x={\left(\frac{3^2}{2^2}\right)}^{-2}$

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

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

$\Rightarrow x=-\frac{2^{3-4}}{3^{3-4}}$ using $\frac{a^m}{a^n}=a^{m-n}$

$\Rightarrow x=-\frac{2^{-1}}{3^{-1}}=\frac{-3}{2}$ using $a^{-m}=\frac{1}{a^m}$