How can I simplify $\sqrt{\frac{5+\sqrt{5}}{2}}$? I've tried to see the root as
$\sqrt{\frac{5+\sqrt{5}}{2}} = \sqrt{a}+\sqrt{b},$
but this method doesn't give me something good.

We can write it so:
$$\sqrt{\frac{5+\sqrt{5}}{2}}=\sqrt{\frac{5+\sqrt{5}}{2}}+\sqrt0.$$
Let there be rationals $a$ and $b$ for which $\sqrt{\frac{5+\sqrt{5}}{2}}=\sqrt{a}+\sqrt{b}.$
Thus, $$\frac{5+\sqrt5}{2}=a+b+2\sqrt{ab},$$ which gives
$$a+b=\frac{5}{2}$$ and
$$ab=\frac{5}{16},$$ which says that $a$ and $b$ are roots of the equation
$$x^2-\frac{5}{2}x+\frac{5}{16}=0,$$ which is a contradiction because it is easy to see that this equation has no rational roots.
