Question

Evaluate: $\frac{2^{1/2}\cdot 3^{1/3}\cdot 4^{1/4}}{{10}^{-1/5}\cdot 5^{3/5}}÷\frac{3^{4/3}\cdot 5^{-7/5}}{4^{-3/5}\cdot 6}$

Answer 1

The given expression

$\begin{array}{cc}& \frac{2^{1/2}\cdot 3^{1/3}\cdot (2{)}^{2/4}}{(2\times 5{)}^{-1/5}\cdot 5^{3/5}}\times \frac{{\left(2^2\right)}^{-3/5}\times 2\times 3}{3^{4/3}\times 5^{-7/5}}\\ & =\frac{2^{(\frac{1}{2}+\frac{1}{2})}\cdot 3^{1/3}}{2^{-1/5}\times 5^{(\frac{3}{5}}-\frac{1}{5})}\times \frac{2^{-6/5}\times 2\times 3}{3^{4/3}\times 5^{-7/5}}\\ & =2^{(1+1/5-6/5+1)}\times 3^{(\frac{1}{3}+1-\frac{4}{3})}\times 5^{(\frac{7}{5}-\frac{2}{5})}\\ & =2^1\times 3^0\times 5^1=10\end{array}$

Hence, the value of given expression is 2.