Question

Find the product of $\sqrt[3]{35}$ and $\sqrt[4]{44}$.

Answer 1

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

$\sqrt[3]{35}\times \sqrt[4]{44}=5^{\frac{1}{3}}\times 4^{\frac{1}{4}}$

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

$\sqrt[3]{35}\times \sqrt[4]{44}=5^{\frac{1}{3}}\times 4^{\frac{1}{4}}=5^{\frac{1}{3}\times \frac{12}{12}}\times 4^{\frac{1}{4}\times \frac{12}{12}}=5^{\frac{4}{12}}\times 4^{\frac{3}{12}}=\sqrt[12]{125^4}\times \sqrt[12]{124^3}=\sqrt[12]{125^4\times 4^3}=\sqrt[12]{12625\times 64}=\sqrt[12]{1240000}$.

Hence, $\sqrt[3]{35}\times \sqrt[4]{44}=\sqrt[12]{1240000}$.