Question

Find the product of $\sqrt[3]{32}$ and $\sqrt[6]{65}$.

Answer 1

The product of the given surds can be determined as follows:

$\sqrt[3]{32}\times \sqrt[6]{65}=2^{\frac{1}{3}}\times 5^{\frac{1}{6}}$

Now, $LCM$ of $3$ and $6$ is $6$, therefore,

$\sqrt[3]{32}\times \sqrt[6]{65}=2^{\frac{1}{3}}\times 5^{\frac{1}{6}}=2^{\frac{1}{3}\times \frac{6}{6}}\times 5^{\frac{1}{6}\times \frac{6}{6}}=2^{\frac{2}{6}}\times 5^{\frac{1}{6}}=\sqrt[6]{62^2}\times \sqrt[6]{65^1}=\sqrt[6]{64\times 5}=\sqrt[6]{620}$.

Hence, $\sqrt[3]{32}\times \sqrt[6]{65}=\sqrt[6]{620}$.