Question

if ${\left(\frac{9}{4}\right)}^{-4}\times {\left(\frac{2}{3}\right)}^3={\left(\frac{p}{q}\right)}^{11}$,then find the value of${\left(\frac{p}{q}\right)}^{-2}$.

Answer 1

${\left(\frac{9}{4}\right)}^{-4}\times {\left(\frac{2}{3}\right)}^3={\left(\frac{p}{q}\right)}^{11}$

$\Rightarrow {\left({\left(\frac{3}{2}\right)}^2\right)}^{-4}\times {\left(\frac{2}{3}\right)}^3={\left(\frac{p}{q}\right)}^{11}$

$\Rightarrow {\left(\frac{3}{2}\right)}^{-8}\times {\left(\frac{3}{2}\right)}^3={\left(\frac{p}{q}\right)}^{11}$

$\Rightarrow {\left(\frac{3}{2}\right)}^{-11}={\left(\frac{p}{q}\right)}^{11}$

$\Rightarrow {\left(\frac{2}{3}\right)}^{11}={\left(\frac{p}{q}\right)}^{11}$

$\Rightarrow \frac{p}{q}=\frac{2}{3}$

${\left(\frac{p}{q}\right)}^{-2}={\left(\frac{2}{3}\right)}^{-2}={\left(\frac{3}{2}\right)}^2=\frac{9}{4}$