<s>[INST] Simplifying radicals inside radicals: $\sqrt{24+8\sqrt{5}}$ Simplify: $\sqrt{24+8\sqrt{5}}$
I removed the common factor 4 out of the square root to obtain $2\sqrt{6+2\sqrt{5}}$, but the answer key says it is $2+2\sqrt{5}$.
Am I missing out on some general rule here?
 [/INST] We can generalize this into a formula, but you can ask me later if you want. For now, we denest $\sqrt{24+8\sqrt{5}}$.
To denest, you have to assume that the radical can be rewritten as the sum of two other radicals (surds). So we have $$\sqrt{24+8\sqrt{5}}=\sqrt{x}+\sqrt{y}$$
Squaring both sides gives us $$24+8\sqrt{5}=x+y+2\sqrt{xy}$$
So we have $x+y=24$ and $2\sqrt{xy}=8\sqrt{5}$. So $x\cdot y=80$. This can be easily solved by finding two numbers whose sum is $24$ and their product is $80$.
When $x=20$ and $y=4$, the conditions are met. So $\sqrt{24+8\sqrt{5}}=\sqrt{20}+\sqrt{4}=2+2\sqrt{5}$.
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