Question

Simplify ${\left(\frac{3125}{243}\right)}^{\frac{4}{5}}$.

Answer 1

$$\begin{gathered} {\left(\frac{3125}{243}\right)}^{\frac{4}{5}} \\ ={\left(\frac{5\times 5\times 5\times 5\times 5}{3\times 3\times 3\times 3\times 3}\right)}^{\frac{4}{5}} \\ ={\left(\frac{5^5}{3^5}\right)}^{\frac{4}{5}} \\ ={\left[{\left(\frac{5}{3}\right)}^5\right]}^{\frac{4}{5}}\left[{\left(\frac{x}{y}\right)}^a=\frac{x^a}{y^a}\right] \end{gathered}$$
$$\begin{gathered} ={\left(\frac{5}{3}\right)}^{5\times \frac{4}{5}}\left[{\left(x^a\right)}^b=x^{ab}\right] \\ ={\left(\frac{5}{3}\right)}^4 \\ =\frac{5\times 5\times 5\times 5}{3\times 3\times 3\times 3} \\ =\frac{625}{81} \end{gathered}$$