Question

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

Answer 1

Orders of the given surds are $2$ and $3$.

Now, LCM of $2$ and $3$ is $6$.
$\sqrt{3}\times \sqrt[3]{35}=3^{\frac{1}{2}}\times 5^{\frac{1}{3}}$ (index form of the surds)
$=3^{\frac{1}{2}\times \frac{6}{6}}\times 5^{\frac{1}{3}\times \frac{6}{6}}$ (each index is multiplied and divided by the LCM)
$=3^{\frac{3}{6}}\times 5^{\frac{2}{6}}=3^{3\times \frac{1}{6}}\times 5^{2\times \frac{1}{6}}$
$=\sqrt[6]{63^3}\times \sqrt[6]{65^2}$ (rewrite the index form into surd form)

$=\sqrt[6]{63^3\times 5^2}=\sqrt[6]{627\times 25}$ (using the rule $\sqrt[n]{na}\times \sqrt[n]{nb}=\sqrt[n]{nab})$
$=\sqrt[6]{6675}$ .

$\therefore \sqrt{3}\times \sqrt[3]{35}=\sqrt[6]{6675}$ .