Question

Thu gọn biểu thức:

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

Answer 1

Ta có: \(E=\frac{1}{\sqrt{2}-\sqrt{3}}\cdot\sqrt{\frac{3\sqrt{2}-2\sqrt{3}}{3\sqrt{2}+2\sqrt{3}}}\)

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

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

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

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

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

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