Question

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

Answer 1

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+2sqrt(ab)+b`
And you put:
`3=a+b`
`sqrt(5)=2sqrt(ab)` that can be squared again giving:
`5=4ab`
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(1/2)+sqrt(5/2)` and
`sqrt(2+sqrt(3))=sqrt(1/2)+sqrt(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