Question

Tính:\(\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)

Answer 1

Có: \(\frac{1}{\sqrt{2}}\left(\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\right)\)

\(=\frac{2+\sqrt{3}}{4+\sqrt{4+2\sqrt{3}}}+\frac{2-\sqrt{3}}{4-\sqrt{4-2\sqrt{3}}}\)

\(=\frac{2+\sqrt{3}}{4+\sqrt{\left(\sqrt{3}+1\right)^2}}+\frac{2-\sqrt{3}}{4-\sqrt{\left(\sqrt{3}-1\right)^2}}\)

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

\(=\frac{2+\sqrt{3}}{5+\sqrt{3}}+\frac{2-\sqrt{3}}{5-\sqrt{3}}\)

\(=\frac{\left(2+\sqrt{3}\right)\left(5-\sqrt{3}\right)+\left(2-\sqrt{3}\right)\left(5+\sqrt{3}\right)}{\left(5+\sqrt{3}\right)\left(5-\sqrt{3}\right)}\)

\(=\frac{10-2\sqrt{3}+5\sqrt{3}-3+10+2\sqrt{3}-5\sqrt{3}-3}{25-3}\)

\(=\frac{14}{22}=\frac{7}{11}\)

Answer 2

nhưng nếu nhập vào máy tính kết quả ra \(\sqrt{2}\)