Question

By what number should ${\left(\frac{-3}{2}\right)}^{-3}$be divided so that the quotient may be ${\left(\frac{4}{27}\right)}^{-2}$?

Answer 1

Let ${\left(\frac{-3}{2}\right)}^{-3}$ be divided by x to get ${\left(\frac{4}{27}\right)}^{-2}$ as quotient,
Then${\left(\frac{-3}{2}\right)}^{-3}÷x={\left(\frac{4}{27}\right)}^{-2}$
$\Rightarrow x={\left(\frac{-3}{2}\right)}^{-3}÷{\left(\frac{2^2}{3^3}\right)}^{-2}={\left(\frac{-3}{2}\right)}^{-3}÷\frac{(2{)}^{-4}}{(3{)}^{-6}}$
$={\left(\frac{-3}{2}\right)}^{-3}\times \frac{(3{)}^{-6}}{(2{)}^{-4}}$
$=\frac{(-3{)}^{-3}\times (3{)}^{-6}}{2^{-3}\times 2^{-4}}=-\frac{3^{-9}}{2^{-7}}$
$\therefore x=-\frac{2^7}{3^9}$