Question

Find the product of $\sqrt[3]{33}$ and $\sqrt[4]{42}$.

Answer 1

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

$\sqrt[3]{33}\times \sqrt[4]{42}=3^{\frac{1}{3}}\times 2^{\frac{1}{4}}$

Now, $LCM$ of $3$ and $4$ is $12$, therefore,

$\sqrt[3]{33}\times \sqrt[4]{42}=3^{\frac{1}{3}}\times 2^{\frac{1}{4}}=3^{\frac{1}{3}\times \frac{12}{12}}\times 2^{\frac{1}{4}\times \frac{12}{12}}=3^{\frac{4}{12}}\times 2^{\frac{3}{12}}=\sqrt[12]{123^4}\times \sqrt[12]{122^3}=\sqrt[12]{123^4\times 2^3}=\sqrt[12]{1281\times 8}=\sqrt[12]{12648}$.

Hence, $\sqrt[3]{33}\times \sqrt[4]{42}=\sqrt[12]{12648}$.