Question

$\frac{2^{\frac{1}{2}}\times 3^{\frac{1}{3}}\times 4^{\frac{1}{4}}}{{10}^{-\frac{1}{5}}\times 5^{\frac{3}{5}}}÷\frac{3^{\frac{4}{3}}\times 5^{-\frac{7}{5}}}{4^{-\frac{3}{5}}\times 6}$

Answer 1

$\frac{2^{\frac{1}{2}}\times 3^{\frac{1}{3}}\times (2{)}^{2\times \frac{1}{4}}}{(2\times 5{)}^{-\frac{1}{5}}\times 5^{\frac{3}{5}}}\times \frac{4^{-\frac{3}{5}}\times 6}{3^{\frac{4}{3}}\times 5^{-\frac{7}{5}}}$
$=\frac{2^{\frac{1}{2}}\times 3^{\frac{1}{3}}\times 2^{\frac{1}{2}}}{2^{-\frac{1}{5}}\times 5^{-\frac{1}{5}}\times 5^{\frac{3}{5}}}\times \frac{(2{)}^{2(-\frac{3}{5})}\times (2\times 3)}{3^{\frac{4}{3}}\times (5{)}^{-\frac{1}{5}}}=[2^{(\frac{1}{2}+\frac{1}{2}-(\frac{-1}{5}\left)\right)}\times 3^{\frac{1}{3}}\times 5-(-\frac{1}{5}+\frac{3}{5}\left)\right]\times [2^{(-\frac{6}{5}+1)}\times 3^{1.\frac{4}{3}}\times 5^{(-\frac{1}{5})}]=(2^{\frac{6}{5}}\times 3^{\frac{1}{3}}\times 5^{-\left(\frac{2}{5}\right)})\times (2^{-\frac{1}{5}}\times 3^{-\frac{1}{3}}\times 5^{\frac{7}{5}})$
$=2^{(\frac{6}{5}-\frac{1}{5})}\times 3^{(\frac{1}{3}-\frac{1}{3})}\times 5^{(-\frac{2}{5}+\frac{7}{5})}=2^{\frac{5}{5}}\times 3^{\circ }\times 5^{\frac{5}{5}}=2^1\times 1\times 5^1=10$