Question

Solve the following equation:
$\sqrt{{\log }_{2}x}+\sqrt[3]{3{\log }_{2}x}=2$

Answer 1

$\sqrt{{log}_{2}x}+\sqrt[3]{3{log}_{2}x}=2$

This will be possible only if ${log}_{2}x=1$

(Therefore, sum of $a^{\frac{1}{2}}$ and $a^{\frac{1}{3}}$ will be rational or Natural number only if $a$ have both square root and cube root.

Therefore, $a$ can be $1$, $64.$

Here $a={log}_{2}x\Rightarrow x=2$