Question

Simplify the following and express with positive index:
${\left(\frac{(27{)}^{-3}}{9^{-3}}\right)}^{\frac{1}{5}}$

Answer 1

Simplifying the given expression
with positive index
Given,
${\left(\frac{(27{)}^{-3}}{9^{-3}}\right)}^{\frac{1}{5}}$
It can be simplified as:
${\left(\frac{(27{)}^{-3}}{9^{-3}}\right)}^{\frac{1}{5}}$
$={\left(\frac{\frac{1}{(27{)}^3}}{\frac{1}{9^3}}\right)}^{\frac{1}{5}}[\because a^{-n}=\frac{1}{a^n}]$
$={\left(\frac{9^3}{{27}^3}\right)}^{\frac{1}{5}}={\left(\left({\frac{9}{27}}^3\right)\right)}^{\frac{1}{5}}[\frac{a^m}{b^m}=\left({\frac{a}{b}}^m\right)]$
On further calculation, we have:
$={\left({\left(\frac{1}{3}\right)}^3\right)}^{\frac{1}{5}}={\left(\frac{1}{3^3}\right)}^{\frac{1}{5}}$
$=\frac{1}{3^{\frac{3}{5}}}$ $[\because (a^m{)}^n=a^{m\times n}]$
$=\frac{1}{3^{\frac{3}{5}}}$

Hence, the simplified value with positive index
is $\frac{1}{3^{\frac{3}{5}}}$.