Question

simplify:

(243/32)^-2/5x (2/3)^-3 x {(125/27)^2/3 +(3/5)^-2 )} divided by {(256)^1/4 x (625)^1/4

Answer 1

$$\begin{gathered} \frac{{\left({\displaystyle \frac{243}{32}}\right)}^{-{\displaystyle \frac{2}{5}}}\times {\left({\displaystyle \frac{2}{3}}\right)}^{-3}\times \left\{{\left({\displaystyle \frac{125}{27}}\right)}^{{\displaystyle \frac{2}{3}}}+{\left({\displaystyle \frac{3}{5}}\right)}^{-2}\right\}}{\left\{{\left(256\right)}^{{\displaystyle \frac{1}{4}}}\times {\left(625\right)}^{{\displaystyle \frac{1}{4}}}\right\}} \\ =\frac{{\left\{{\left({\displaystyle \frac{3}{2}}\right)}^5\right\}}^{-{\displaystyle \frac{2}{5}}}\times {\left({\displaystyle \frac{2}{3}}\right)}^{-3}\times \left[{\left\{{\left({\displaystyle \frac{5}{3}}\right)}^3\right\}}^{{\displaystyle \frac{2}{3}}}+{\left({\displaystyle \frac{5}{3}}\right)}^2\right]}{{\left\{{\left(4\right)}^4\right\}}^{{\displaystyle \frac{1}{4}}}\times {\left\{{\left(5\right)}^4\right\}}^{{\displaystyle \frac{1}{4}}}} \\ =\frac{{\left({\displaystyle \frac{3}{2}}\right)}^{-2}\times {\left({\displaystyle \frac{2}{3}}\right)}^{-3}\times \left[{\left({\displaystyle \frac{5}{3}}\right)}^2+{\left({\displaystyle \frac{5}{3}}\right)}^2\right]}{4\times 5} \\ =\frac{{\left({\displaystyle \frac{2}{3}}\right)}^2\times {\left({\displaystyle \frac{3}{2}}\right)}^3\times \left[{\displaystyle \frac{25}{9}}+{\displaystyle \frac{25}{9}}\right]}{20} \\ =\frac{{\displaystyle \frac{4}{9}}\times {\displaystyle \frac{27}{8}}\times \left[{\displaystyle \frac{50}{9}}\right]}{20} \\ =\frac{{\displaystyle \frac{3}{2}}\times {\displaystyle \frac{50}{9}}}{20} \\ =\frac{{\displaystyle \frac{25}{3}}}{20}=\frac{25}{3}\times \frac{1}{20}=\frac{5}{12} \end{gathered}$$