Question

Chứng minh đẳng thức \(\sqrt{5-2\sqrt{6}}-\sqrt{5+\sqrt{6}}=-2\sqrt{2}\)

Answer 1

Sửa đề: \(\sqrt{5-2\sqrt{6}}-\sqrt{5+2\sqrt{6}}\)

Ta có: \(\sqrt{5-2\sqrt{6}}-\sqrt{5+2\sqrt{6}}\)

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

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

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

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

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

\(=-2\sqrt{2}\)(đpcm)