# How do you solve sqrt(3 + sqrt5) -sqrt(2 + sqrt3)?

Jan 7, 2015

Here I can give you only a hint (I have to go to work!).

I would use a technique called "denesting";
You can write:
$\sqrt{3 + \sqrt{5}} = \sqrt{a} + \sqrt{b}$
You square both sides:
$3 + \sqrt{5} = a + 2 \sqrt{a b} + b$
And you put:
$3 = a + b$
$\sqrt{5} = 2 \sqrt{a b}$ that can be squared again giving:
$5 = 4 a b$
You must now solve this system in $a$ and $b$ (for both square roots in your original expression).
You should get:
$\sqrt{3 + \sqrt{5}} = \sqrt{\frac{1}{2}} + \sqrt{\frac{5}{2}}$ and
$\sqrt{2 + \sqrt{3}} = \sqrt{\frac{1}{2}} + \sqrt{\frac{3}{2}}$.
Try substituring in your original expression and eventually rationalize.

PLEASE check my numbers and operations because I did them in a hurry!!! :-l