1395.21 – Radical in Radical

OK: $\sqrt{5+ \sqrt{24}} = \sqrt{a} + \sqrt{b}$, where $a$ and $b$ are integers. Then,

$$\begin{align*}5 + \sqrt{24} &= 5+2\sqrt{6} \\&= (\sqrt{a} + \sqrt{b})^2 \\&=a+2\sqrt{ab}+b\end{align*}$$

So $a+b=5$, and $\sqrt{ab}=\sqrt{6}$. Therefore $a = 2$ and $b = 3$ or vice versa. So $\sqrt{5+ \sqrt{24}} =\sqrt{2} +\sqrt{3}$. Who'd a thunk it!