simplify 2$\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}$

Question

Simplify the following expression:

2$\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}$

I would be grateful to get a full response.

Answer 1

Since $13+4\sqrt{3}=(1+2\sqrt{3})^2$ and $4-2\sqrt{3}=(\sqrt{3}-1)^2$, your surd is$$2\sqrt{2+\sqrt{3}}=\sqrt{8+2\sqrt{12}}=\sqrt{2}+\sqrt{6}.$$

Answer 2

Hint

$$13+\sqrt{48}=12+1+2\sqrt{12}=(\sqrt{12}+1)^2$$

Alternatively

We need $$a^2+b^2=13,a^2b^2=12$$ where $a,b>0$

So, $a^2,b^2$ are the roots of $$t^2-13t+12=0$$