Question

Simplify the following ${\left[{\left\{{\left(625\right)}^{\frac{-1}{2}}\right\}}^{\frac{-1}{4}}\right]}^2$

Answer 1

Simplify the expression:

Given ${\left[{\left\{{\left(625\right)}^{\frac{-1}{2}}\right\}}^{\frac{-1}{4}}\right]}^2$

We know that

${\left(a^m\right)}^{\mathbf{n}}\mathbf{=}{\mathit{a}}^{\mathbf{m}\mathbf{n}}$

Consider,

${\left[{\left\{{\left(625\right)}^{\frac{-1}{2}}\right\}}^{\frac{-1}{4}}\right]}^2$

$$\begin{gathered} ={\left\{{\left(625\right)}^{\frac{-1}{2}}\right\}}^{\frac{-2}{4}} \\ ={\left(625\right)}^{\frac{-1\times -2}{2\times 4}} \\ ={\left(625\right)}^{\frac{1}{4}} \\ ={\left(5^4\right)}^{\frac{1}{4}} \\ =5^{4\times \frac{1}{4}} \\ =5 \end{gathered}$$

Hence, ${\left[{\left\{{\left(625\right)}^{\frac{-1}{2}}\right\}}^{\frac{-1}{4}}\right]}^2$ is $5$