Question

Represent the following mixed infinite decimal periodic fractions as common fractions $\left(14\right)$
$\frac{\sqrt[3]{3a^5{b}^{1/2}}\sqrt[4]{4a^{-1}}}{(a^2\sqrt[5]{5a{b}^3}{)}^2}$

Answer 1

The given equation is

$\frac{\sqrt[3]{3a^5{b}^{1/2}}\sqrt[4]{4a^{-1}}}{{\left(a^2\sqrt[5]{5a{b}^3}\right)}^2}$

$\Rightarrow =\frac{a^{5/3}{b}^{1/6}{a}^{-1/4}}{{(a}^2{a}^{1/5}{b}^{3/5}}$[since ${\left(a^m\right)}^n=a^{mn}$]

$\Rightarrow \frac{a^{5/3}{b}^{1/6}{a}^{-1/4}}{{(a}^2{a}^{1/5}{b}^{3/5}}=a^{5/3-2-1/5-1/4}{b}^{1/6-3/5}=a^{-47/60}{b}^{-13/30}$