Question

The value of the expression $\frac{1}{1+\sqrt{2}+\sqrt{3}}+\frac{1}{1-\sqrt{2}+\sqrt{3}}+\frac{1}{1+\sqrt{2}-\sqrt{3}}+\frac{1}{1-\sqrt{2}-\sqrt{3}}$, is

Answer 1

Answer: 2

$\frac{1}{1+(\sqrt{2}+\sqrt{3})}+\frac{1}{1-(\sqrt{2}+\sqrt{3})}+\frac{1}{1-(\sqrt{2}-\sqrt{3})}+\frac{1}{1+(\sqrt{2}-\sqrt{3})}$
$=\frac{1-(\sqrt{2}+\sqrt{3})+1+\sqrt{2}+\sqrt{3}}{1-(\sqrt{2}+\sqrt{3}{)}^2}+\frac{1+\sqrt{2}-\sqrt{3}+1-\sqrt{2}+\sqrt{3}}{1-(\sqrt{2}-\sqrt{3}{)}^2}$
$=\frac{2}{-4-2\sqrt{6}}+\frac{2}{-4+2\sqrt{6}}=\frac{1}{-2-\sqrt{6}}+\frac{1}{-2+\sqrt{6}}=\frac{-2+\sqrt{6}-2-\sqrt{6}}{4-6}=\frac{-4}{-2}=2$