<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] $$\sqrt{24+8\sqrt{5}}=\sqrt{20+4+2\cdot 2\cdot 2\sqrt{5}}=\sqrt{(2\sqrt5)^2+2^2+2 \cdot 2 \cdot 2\sqrt{5}}=\sqrt{(2+2\sqrt5)^2}=2+2\sqrt5$$
or, to continue on your start,
$$\sqrt{6+2\sqrt5}=\sqrt{1^2+(\sqrt5)^2+2\sqrt5}=1+\sqrt5$$
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