Question

Find the product of $\sqrt{3}$ and $\sqrt[4]{45}$.

Answer 1

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

$\sqrt{3}\times \sqrt[4]{45}=3^{\frac{1}{2}}\times 5^{\frac{1}{4}}$

$LCM$ of $2$ and $4$ is $4$, therefore,

$\sqrt{3}\times \sqrt[4]{45}=3^{\frac{1}{2}}\times 5^{\frac{1}{4}}=3^{\frac{1}{2}\times \frac{4}{4}}\times 5^{\frac{1}{4}\times \frac{4}{4}}=3^{\frac{2}{4}}\times 5^{\frac{1}{4}}=\sqrt[4]{43^2}\times \sqrt[4]{45^1}=\sqrt[4]{43^2\times 5}=\sqrt[4]{49\times 5}=\sqrt[4]{445}$.

Hence, $\sqrt{3}\times \sqrt[4]{45}=\sqrt[4]{445}$.