Question

Find the value of the following expression:

${625}^{-3/4}$

Answer 1

Step 1: Factorise $625$ into prime numbers.

$\begin{array}{l}{625}^{-3/4}\\ ={\left(5\times 5\times 5\times 5\right)}^{-3/4}\\ ={\left(5^4\right)}^{-3/4}\end{array}$

Step 2: Simplify the above expression using the laws of rational exponents.

$\begin{array}{l}{\left(5^4\right)}^{-3/4}\\ =\frac{1}{{\left(5^4\right)}^{3/4}}\left[\because p^{-n}=\frac{1}{p^n}\right]\\ =\frac{1}{{\left(5\right)}^{4\times \frac{3}{4}}}\left[\because {\left(p^n\right)}^m=p^{m\times n}\right]\\ =\frac{1}{{\left(5\right)}^3}\\ =\frac{1}{5\times 5\times 5}\\ =\frac{1}{125}\end{array}$

Hence, the value of the given expression is $\frac{1}{125}$.