Prove: \[ \left(\frac{\sqrt{3\cdot 5^{-3}}}{\sqrt[3]{3^{-1}\cdot 5}}\right)\times \sqrt[6]{3\cdot 5^6} = \frac{3}{5} \]
Proof
\[ = \frac{(3\cdot 5^{-3})^{1/2}}{(3^{-1}\cdot 5)^{1/3}} \times (3\cdot 5^6)^{1/6} \]
\[ = \frac{3^{1/2} \cdot 5^{-3/2}}{3^{-1/3}\cdot 5^{1/3}} \times 3^{1/6}\cdot 5 \]
\[ = 3^{1/2 + 1/3 + 1/6} \cdot 5^{-3/2 – 1/3 + 1} \]
\[ = 3^{\frac{3+2+1}{6}} \cdot 5^{\frac{-9-2+6}{6}} \]
\[ = 3^1 \cdot 5^{-5/6} \]
\[ = \frac{3}{5} \]