Question

Find the value of the following expression:

$\frac{8^{-2/3}}{{16}^{-3/4}}$

Answer 1

Step 1: Factorise $8\text{and 16}$ into prime numbers.

$\begin{array}{l}\frac{8^{-2/3}}{{16}^{-3/4}}\\ =\frac{{\left(8\right)}^{-2/3}}{{\left(16\right)}^{-3/4}}\\ =\frac{{\left(2\times 2\times 2\right)}^{-2/3}}{{\left(2\times 2\times 2\times 2\right)}^{-3/4}}\\ =\frac{{\left(2^3\right)}^{-2/3}}{{\left(2^4\right)}^{-3/4}}\end{array}$

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

$\begin{array}{l}\frac{{\left(2^3\right)}^{-2/3}}{{\left(2^4\right)}^{-3/4}}\\ =\frac{{\left(2^4\right)}^{3/4}}{{\left(2^3\right)}^{2/3}}\left[\because p^{-n}=\frac{1}{p^n}\right]\\ =\frac{{\left(2\right)}^{4\times \frac{3}{4}}}{{\left(2\right)}^{3\times \frac{2}{3}}}\left[\because {\left(p^n\right)}^m=p^{m\times n}\right]\\ =\frac{{\left(2\right)}^3}{{\left(2\right)}^2}\\ ={\left(2\right)}^{3-2}\left[\because \frac{a^m}{a^n}=a^{m-n}\right]\\ =2\end{array}$

Hence, the value of the given expression is $2$.