evaluate $\frac 1{1+\sqrt2+\sqrt3} + \frac 1{1-\sqrt2+\sqrt3} + \frac 1{1+\sqrt2-\sqrt3} + \frac 1{1-\sqrt2-\sqrt3}$ 
Evaluate $\frac 1{1+\sqrt2+\sqrt3} + \frac 1{1-\sqrt2+\sqrt3} + \frac 1{1+\sqrt2-\sqrt3} + \frac 1{1-\sqrt2-\sqrt3}$

How to evalute this equation without using calculator?

We have that $\pm\sqrt{2}\pm\sqrt{3}$ are the roots of the polynomial $(x^2-5)^2-24$, hence:
$$ x^4-10\,x^2+1 = \prod_{\xi_i\in Z}(x-\xi) $$
and $1\pm\sqrt{2}\pm\sqrt{3}$ are the roots of the polynomial:
$$ (x-1)^4-10(x-1)^2+1 = x^4-4x^3-4x^2+16x-8. $$
By Viète's theorem, the sum of the roots of a polynomial $p(x)$ raised to the minus one power is given by $-\frac{[x^1]\,p(x)}{[x^0]\,p(x)}$, hence the answer is $-\frac{16}{-8} = \color{red}{2}$.
