Question

Reduce the following to surd of the same order:
$\sqrt{5},\sqrt[4]{415}$ and $\sqrt[8]{850}$

Answer 1

The given surds $\sqrt{5},\sqrt[4]{415},\sqrt[8]{850}$can be rewritten as follows:

$\sqrt{5}=5^{\frac{1}{2}}\sqrt[4]{415}={15}^{\frac{1}{4}}\sqrt[8]{850}={50}^{\frac{1}{8}}$

The denominators of the powers of above surds are $2,4$ and $8$ and the $LCM$ of $2,4$ and $8$ is $8$. Therefore,

$\sqrt{5}=5^{\frac{1}{2}}=5^{\frac{1\times 4}{2\times 4}}=5^{\frac{4}{8}}=\sqrt[8]{85^4}=\sqrt[8]{8625}\sqrt[4]{415}={15}^{\frac{1}{4}}={15}^{\frac{1\times 2}{4\times 2}}={15}^{\frac{2}{8}}=\sqrt[8]{8{15}^2}=\sqrt[8]{8225}\sqrt[8]{850}$

Hence, the surds $\sqrt{5},\sqrt[4]{415},\sqrt[8]{850}$ are reduced to the surd of same order $8$.