Question

Find the product of $\sqrt{5}$ and $\sqrt[3]{33}$.

Answer 1

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

$\sqrt{5}\times \sqrt[3]{33}=5^{\frac{1}{2}}\times 3^{\frac{1}{3}}$

Now, $LCM$ of $2$ and $3$ is $6$, therefore,

$\sqrt{5}\times \sqrt[3]{33}=5^{\frac{1}{2}}\times 3^{\frac{1}{3}}=5^{\frac{1}{2}\times \frac{6}{6}}\times 3^{\frac{1}{3}\times \frac{6}{6}}=5^{\frac{3}{6}}\times 3^{\frac{2}{6}}=\sqrt[6]{65^3}\times \sqrt[6]{63^2}=\sqrt[6]{65^3\times 3^2}=\sqrt[6]{6125\times 9}=\sqrt[6]{61125}$.

Hence, $\sqrt{5}\times \sqrt[3]{33}=\sqrt[6]{61125}$.