question_answer

  By what number should \[{{\left( -\frac{3}{2} \right)}^{-3}}\]  be divided so that the quotient is\[{{\left( \frac{9}{4} \right)}^{-2}}\]?

Answer:

Let the required number = x

According to problem,

\[{{\left( \frac{-3}{2} \right)}^{-3}}\div x={{\left( \frac{9}{4} \right)}^{-2}}\]

or         \[{{\left( \frac{2}{-3} \right)}^{-3}}\div x={{\left( \frac{4}{9} \right)}^{2}}\]

or          \[\frac{8}{-27}\times {{x}^{-1}}=\frac{16}{81}\] \[\left[ \because \,\,\frac{1}{{{x}^{a}}}={{x}^{-a}} \right]\]

or           \[{{x}^{-1}}=\frac{16}{81}\times \frac{-27}{8}\]

or       \[{{x}^{-1}}=\frac{-2}{3}\]

or        \[\frac{1}{x}=\frac{-2}{3}\]

or         \[x=\frac{-3}{2}\]

Hence,   required number \[=\frac{-3}{2}\]