Question

Find the product of $\sqrt[3]{32}$ and $\sqrt[4]{43}$.

Answer 1

Orders of the given surds are $3$ and $4$.

Now, LCM of $3$ and $4$ is $12$.
$\sqrt[3]{32}\times \sqrt[4]{43}=2^{\frac{1}{3}}\times 3^{\frac{1}{4}}$ (index form of the surds)
$=2^{\frac{1}{3}\times \frac{12}{12}}\times 3^{\frac{1}{4}\times \frac{12}{12}}$ (each index is multiplied and divided by the LCM)
$=2^{\frac{4}{12}}\times 3^{\frac{3}{12}}=2^{4\times \frac{1}{12}}\times 3^{3\times \frac{1}{12}}$
$=\sqrt[12]{122^4}\times \sqrt[12]{123^3}$ (rewrite the index form into surd form)

$=\sqrt[12]{122^4\times 3^3}=\sqrt[12]{1216\times 27}$ (using the rule $\sqrt[n]{na}\times \sqrt[n]{nb}=\sqrt[n]{nab})$
$=\sqrt[12]{12432}$ .

$\therefore \sqrt[3]{32}\times \sqrt[4]{43}=\sqrt[12]{12432}$ .