OK: $\sqrt{5+ \sqrt24} = \sqrt{a} + \sqrt{b}$, where $a$ and $b$ are integers.

Then $5 + \sqrt{24} =5+2\sqrt{6} = (\sqrt{a} + \sqrt{b})^2=a+2\sqrt{ab}+b.$

So $a+b=5$, and $\sqrt{ab}=\sqrt{6}.$

Therefore $a = 2$ and $b = 3$ or vice versa.

So $\sqrt{5+ \sqrt24} =\sqrt{2} +\sqrt{3}.$ Who'd a thunk it!