Question

\(A=\left(\sqrt{x-4\sqrt{2}}-\sqrt{x+4\sqrt{2}}\right)\sqrt{x+\sqrt{x^2-32}}\) với \(x\ge4\sqrt{2}\)

Answer 1

Lời giải:

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

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

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

\(=(\sqrt{x-4\sqrt{2}})^2-(\sqrt{x+4\sqrt{2}})^2=(x-4\sqrt{2})-(x+4\sqrt{2})=-8\sqrt{2}\)

Answer 2

Lời giải:

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

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

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

\(=(\sqrt{x-4\sqrt{2}})^2-(\sqrt{x+4\sqrt{2}})^2=(x-4\sqrt{2})-(x+4\sqrt{2})=-8\sqrt{2}\)