Question

Simplify the following and express with positive index:

${\left(\frac{3^{-4}}{2^{-8}}\right)}^{\frac{1}{4}}$

Answer 1

Simplifying the given expression with positive index

${\left(\frac{3^{-4}}{2^{-8}}\right)}^{\frac{1}{4}}$

Given,${\left(\frac{3^{-4}}{2^{-8}}\right)}^{\frac{1}{4}}$

It can be simplified as :

${\left(\frac{3^{-4}}{2^{-8}}\right)}^{\frac{1}{4}}$=$[\therefore a^{-1}=\frac{1}{a}]$

${\left(\frac{2^8}{3^4}\right)}^{\frac{1}{4}}$ =$\frac{(2^8{)}^{\frac{1}{4}}}{(3^4{)}^{\frac{1}{4}}}$

$\frac{2^8\times \frac{1}{4}}{3^4\times \frac{1}{4}}=\frac{2^2}{3^1}=\frac{4}{3}$
$[\therefore (a^m{)}^n=a{{}^m}^{\times }{}^n]$

Hence, simplified value of ${\left(\frac{3^{-4}}{2^{-8}}\right)}^{\frac{1}{4}}$ is $\frac{4}{3}$.