Prove: \[ \frac{2^{1/2}\cdot 3^{1/2}\cdot 4^{1/4}}{10^{-1/5}\cdot 5^{3/5}} \div \frac{3^{4/3}\cdot 5^{-7/5}}{4^{-3/5}\cdot 6} = 10 \]
Solution
\[ = \frac{2^{1/2}\cdot 3^{1/2}\cdot (2^2)^{1/4}}{(2\cdot5)^{-1/5}\cdot 5^{3/5}} \div \frac{3^{4/3}\cdot 5^{-7/5}}{(2^2)^{-3/5}\cdot 2\cdot 3} \]
\[ = \frac{2^{1/2}\cdot 3^{1/2}\cdot 2^{1/2}}{2^{-1/5}\cdot 5^{2/5}} \times \frac{2^{-6/5}\cdot 2\cdot 3}{3^{4/3}\cdot 5^{-7/5}} \]
\[ = 2^{1+1/5-6/5+1} \cdot 3^{1/2+1-4/3} \cdot 5^{-2/5+7/5} \]
\[ = 2^1 \cdot 3^0 \cdot 5^1 \]
\[ = 10 \]