Question

Rút gọn: \(\frac{2}{1+\sqrt{2}-\sqrt{3}}\)

Answer 1

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

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

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

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

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

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

\(=\frac{\sqrt{2}\left(1+\sqrt{2}+\sqrt{3}\right)}{\sqrt{2}.\sqrt{2}}\)

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

Answer 2

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

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

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

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

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