Question

2+√5/√2+√3+√5 +2-√5/√2-√3-√5

Answer 1

Ta có: \(\frac{2+\sqrt5}{\sqrt2+\sqrt3+\sqrt5}+\frac{2-\sqrt5}{\sqrt2-\sqrt3-\sqrt5}\)

\(=\frac{\left(2+\sqrt5\right)\left(\sqrt2-\sqrt3-\sqrt5\right)+\left(2-\sqrt5\right)\left(\sqrt2+\sqrt3+\sqrt5\right)}{2-\left(\sqrt3+\sqrt5\right)^2}\)

\(=\frac{2\sqrt2-2\sqrt3-2\sqrt5+\sqrt{10}-\sqrt{15}-5+2\sqrt2+2\sqrt3+2\sqrt5-\sqrt{10}-\sqrt{15}-5}{2-\left(8+2\sqrt{15}\right)}\)

\(=\frac{4\sqrt2-2\sqrt{15}-10}{-6-2\sqrt{15}}=\frac{-4\sqrt2+2\sqrt{15}+10}{6+2\sqrt{15}}=\frac{-2\sqrt2+\sqrt{15}+5}{3+\sqrt{15}}\)