Simplify $\sqrt{10 + \sqrt{24} + \sqrt{40} + \sqrt{60}} $ Simplify
$$ \sqrt{10 + \sqrt{24} + \sqrt{40} + \sqrt{60}} $$

Attempt:
$$ \sqrt{10 + \sqrt{24} + \sqrt{40} + \sqrt{60}}  =  \sqrt{10 + 2\sqrt{6} + 2\sqrt{10} + 2\sqrt{15}} = \sqrt{10 + 2(\sqrt{6} + \sqrt{10} + \sqrt{15})} $$
let $X =\sqrt{10 + \sqrt{24} + \sqrt{40} + \sqrt{60}}$, then
$$ X^{2} = 10 + 2(\sqrt{6} + \sqrt{10} + \sqrt{15}) $$
How to continue?

Let $$(\sqrt{x}+\sqrt{y}+\sqrt{z})^2=x+y+z+2\sqrt{xy}+2\sqrt{yz}+2\sqrt{xz}$$
So $$\sqrt{x+y+z+2\sqrt{xy}+2\sqrt{yz}+2\sqrt{xz}}=\sqrt{x}+\sqrt{y}+\sqrt{z}$$For $$\sqrt{10+\sqrt{24}+\sqrt{40}+\sqrt{60}}=\sqrt{x}+\sqrt{y}+\sqrt{z}.$$We get $x+y+z=10, 2\sqrt{xy}=\sqrt{24}, 2\sqrt{yz}=\sqrt{40}, 2 \sqrt{xz}=\sqrt{60}.$ So we get $xy=6, yz=10, zx=15 \Rightarrow xyz=30, x=2,y=3, z=5.$ 
Hence the requiredsquare root is $$\sqrt{2}+\sqrt{3}+\sqrt{5}.$$
