<s>[INST] Fully simplifying $\sqrt{13+2\left(\sqrt{2}-\sqrt{10}-\sqrt{20}\right)}$ I have this statement:

Simplify this: $\sqrt{13+2\left(\sqrt{2}-\sqrt{10}-\sqrt{20}\right)}$

I know how to simplify roots like $\sqrt{a \pm k\sqrt{b}}$, But i don't know how to simplify this.
Any hint for an elegant solution is appreciated.
 [/INST] Notice that $$(-\sqrt{x}+\sqrt{y}+\sqrt{z})^2=x+y+z-2\sqrt{xy}+2\sqrt{yz}-2\sqrt{xz}$$
So $$\sqrt{x+y+z-2\sqrt{xy}+2\sqrt{yz}-2\sqrt{xz}}=-\sqrt{x}+\sqrt{y}+\sqrt{z}$$
In your case on comparision you have $x+y+z=13$, $xy=20, zx=10, yz=2.$ So you get $xyz=20$, then $x=10,y=2,z=1$, Hence the answer is $\sqrt{10} -\sqrt{2}- \sqrt{1}.$
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