Question

Rút gọn biểu thức

A= \(\sqrt{x+2\sqrt{2x-4}}-\sqrt{x-2\sqrt{2x-4}}\)

Answer 1

ĐKXĐ: \(x\ge2\)

Ta có: \(A=\sqrt{x+2\sqrt{2x-4}}-\sqrt{x-2\sqrt{2x-4}}\)

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

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

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

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

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

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