Question

Find the product of $\sqrt[3]{34}$ and $\sqrt[5]{52}$.

Answer 1

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

$\sqrt[3]{34}\times \sqrt[5]{52}=4^{\frac{1}{3}}\times 2^{\frac{1}{5}}$

Now, $LCM$ of $3$ and $5$ is $15$, therefore,

$\sqrt[3]{34}\times \sqrt[5]{52}=4^{\frac{1}{3}}\times 2^{\frac{1}{5}}=4^{\frac{1}{3}\times \frac{15}{15}}\times 2^{\frac{1}{5}\times \frac{15}{15}}=4^{\frac{5}{15}}\times 2^{\frac{3}{15}}=\sqrt[15]{154^5}\times \sqrt[15]{152^3}=\sqrt[15]{154^5\times 2^3}=\sqrt[15]{151024\times 8}=\sqrt[15]{158192}$.

Hence, $\sqrt[3]{34}\times \sqrt[5]{52}=\sqrt[15]{158192}$.