Question

Simplify: $\frac{{\left(5^{-3}\right)}^2\times {\left(3^4\right)}^{}}{{\left(3^{-2}\right)}^{-3}\times {\left(5^3\right)}^{-2}}$

Answer 1

The given expression $\frac{(5^{-3}{)}^2\times 3^4}{(3^{-2}{)}^{-3}\times (5^3{)}^{-2}}$ can be simplified as follows:

$\frac{(5^{-3}{)}^2\times 3^4}{(3^{-2}{)}^{-3}\times (5^3{)}^{-2}}=\frac{5^{-6}\times 3^4}{3^6\times 5^{-6}}(\because {\left(x^m\right)}^n=x^{m\times n})=\frac{5^{-6}}{5^{-6}}\times \frac{3^4}{3^6}=5^{-6-(-6)}\times 3^{4-6}(\because \frac{x^m}{x^n}=x^{m-n})$

$=5^{-6+6}\times 3^{-2}=5^0\times \frac{1}{3^2}(\because \frac{1}{x}=x^{-1})=1\times \frac{1}{3^2}(\because x^0=1)=\frac{1}{9}$

Hence $\frac{(5^{-3}{)}^2\times 3^4}{(3^{-2}{)}^{-3}\times (5^3{)}^{-2}}=\frac{1}{9}$.