<s>[INST] 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?
 [/INST] We have, $$\underbrace{\frac {1}{1+\sqrt2+\sqrt3} + \frac {1}{1-\sqrt2+\sqrt3}} + \underbrace{\frac {1}{1+\sqrt2-\sqrt3} + \frac {1}{1-\sqrt2-\sqrt3}}$$ $$=\left(\frac {1}{1+\sqrt2+\sqrt3} + \frac {1}{1-\sqrt2+\sqrt3}\right) +\left( \frac {1}{1+\sqrt2-\sqrt3} + \frac {1}{1-\sqrt2-\sqrt3}\right)$$ $$=\left(\frac {1-\sqrt2+\sqrt3+1+\sqrt2+\sqrt3}{(1+\sqrt2+\sqrt3)(1-\sqrt2+\sqrt3)} \right) +\left( \frac {1-\sqrt2-\sqrt3+1+\sqrt2-\sqrt3}{(1+\sqrt2-\sqrt3)(1-\sqrt2-\sqrt3)} \right)$$ $$=\left(\frac {2+2\sqrt3}{(1+\sqrt3)^2-(\sqrt{2})^2} \right) +\left( \frac {2-2\sqrt{3}}{(1-\sqrt3)^2-(\sqrt{2})^2} \right)$$   $$=\frac{2+2\sqrt{3}}{1+3+2\sqrt{3}-2}+\frac{2-2\sqrt{3}}{1+3-2\sqrt{3}-2}$$ $$=\frac{2+2\sqrt{3}}{2+2\sqrt{3}}+\frac{2-2\sqrt{3}}{2-2\sqrt{3}}$$ $$=1+1=2$$
</s>