Question

Arrange the following surds in ascending order of their magnitudes.
$\sqrt{5},\sqrt[3]{311},2\sqrt[6]{63}$.

Answer 1

L.C.M of $2,3$ and $6$ is $6.$
$\sqrt{5}=\sqrt[2]{25}=\sqrt[2\times 3]{2\times 35^3}=\sqrt[6]{65^3}=\sqrt[6]{6125}$

$\sqrt[3]{311}=\sqrt[3\times 2]{3\times 2{11}^2}=\sqrt[6]{6121}$

$2\sqrt[6]{63}=\sqrt[6]{62^6}\sqrt[6]{63}=\sqrt[6]{62^6\times 3}=\sqrt[6]{664\times 3}=\sqrt[6]{6192}$

Since, $121<125<192$
$\therefore \sqrt[6]{6121}<\sqrt[6]{6125}<\sqrt[6]{6192}$
$\therefore \sqrt[3]{311}<\sqrt{5}<2\sqrt[6]{63}$