Question

Simplify the following and express with positive index:
$(32{)}^{-\frac{2}{5}}÷(125{)}^{-\frac{2}{3}}$

Answer 1

Simplifying the given expression
with positive index
Given
$(32{)}^{-\frac{2}{5}}÷(125{)}^{-\frac{2}{3}}$
It can be simplified as:

$(32{)}^{-\frac{2}{5}}÷(125{)}^{-\frac{2}{3}}$
$=\frac{1}{(32{)}^{\frac{2}{5}}}\times \frac{1}{(125{)}^{\frac{-2}{3}}}$
$[\because a÷b=a\times \frac{1}{b}]$
$=\frac{1}{(2\times 2\times 2\times 2\times 2{)}^{\frac{2}{5}}}\times (125{)}^{\frac{2}{3}}$
$[\because a^{-1}=\frac{1}{a}$]

$=\frac{1}{(2^5{)}^{\frac{2}{5}}}\times (5^3{)}^{\frac{2}{3}}$
$=\frac{5^{3\times \frac{2}{3}}}{2^{5\times \frac{2}{5}}}$ $[\because (a^m{)}^n=a^{m\times n}]$
$=\frac{5^2}{2^2}=\frac{25}{4}or{\left(\frac{5}{2}\right)}^2$

Hence, the simplified value with positive index is
${\left(\frac{5}{2}\right)}^2=\frac{25}{4}$.