Question

Rút gọn:

\(A=\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)

Answer 1

\(A=\dfrac{\left(2+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{2+\sqrt{3}}\right)}{2-2-\sqrt{3}}+\dfrac{\left(2-\sqrt{3}\right)\left(\sqrt{2}+\sqrt{2-\sqrt{3}}\right)}{2-2+\sqrt{3}}\)

\(=\dfrac{-2\sqrt{2}+\sqrt{2}\left(\sqrt{3}+1\right)-\sqrt{6}+\sqrt{6+3\sqrt{3}}}{\sqrt{3}}+\dfrac{2\sqrt{2}+\sqrt{2}\left(\sqrt{3}-1\right)-\sqrt{6}-\sqrt{6-3\sqrt{3}}}{\sqrt{3}}\)

\(=\dfrac{\sqrt{6}+\sqrt{2}-\sqrt{6}+\sqrt{6+3\sqrt{3}}+\sqrt{6}-\sqrt{2}-\sqrt{6}-\sqrt{6-3\sqrt{3}}}{\sqrt{3}}\)

\(=\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\)