Simplify the following and express with positive index :

$\Big(\dfrac{3^{-4}}{2^{-8}}\Big)^{\dfrac{1}{4}}$

Simplifying the expression :

$\Rightarrow \Big(\dfrac{3^{-4}}{2^{-8}}\Big)^{\dfrac{1}{4}} = \dfrac{3^{-4\times \dfrac{1}{4}}}{2^{-8 \times \dfrac{1}{4}}} \\[1em] = \dfrac{3^{-1}}{2^{-2}} = \dfrac{\dfrac{1}{3}}{\dfrac{1}{2^2}} \\[1em] = \dfrac{2^2}{3} = \dfrac{4}{3}.$

Hence, $\Big(\dfrac{3^{-4}}{2^{-8}}\Big)^{\dfrac{1}{4}} = \dfrac{4}{3}$.