# 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.

It can't give you anything good.

Say you have $$\sqrt{a+\sqrt{b}}=\sqrt{x}+\sqrt{y}$$, all numbers rational and $$b$$ not a rational square. Then

$$a+\sqrt{b}=(\sqrt{x}+\sqrt{y})^2=(x+y)+2\sqrt{xy}$$

The uniqueness of quadratic surds forces $$a=x+y,b=4xy$$. Then

$$a^2-b=(x+y)^2-4xy=(x-y)^2,\therefore\text{ a rational square.}$$

But here, $$a=5/2$$ and $$b=5/4$$, giving $$a^2-b=5$$. So we're having a bad day.