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

Question

Simplify

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

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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?

Answer 1

From $6 = 2\times 3$, $10 = 2\times 5$, $15 = 3 \times 5$, observe that $10 + 2(\sqrt6 + 2\sqrt{10} + 2\sqrt{15}) = (\sqrt2+\sqrt3+\sqrt5)^2$.

Answer 2

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 $$\sqrt{2}+\sqrt{3}+\sqrt{5}.$$

Answer 3

You can continue so: $$x^2=10 + 2(\sqrt{6} + \sqrt{10} + \sqrt{15})=2+5+3+ 2(\sqrt{6} + \sqrt{10} + \sqrt{15})=(\sqrt2+\sqrt5+\sqrt3)^2.$$ Can you end it now?