Question

Rút gọn biểu thức:

\(A=\frac{\sqrt{x+4\sqrt{x-4}}+\sqrt{x-4\sqrt{x-4}}}{\sqrt{\frac{16}{x^2}-\frac{8}{x}+1}}\) (với x > 4)

Answer 1

\(A=\frac{\sqrt{x-4+4\sqrt{x-4}+4}+\sqrt{x-4-4\sqrt{x-4}+4}}{\sqrt{\left(\frac{4}{x}-1\right)^2}}\)

\(=\frac{\sqrt{\left(\sqrt{x-4}+2\right)^2}+\sqrt{\left(\sqrt{x-4}-2\right)^2}}{\sqrt{\left(1-\frac{4}{x}\right)^2}}=\frac{\sqrt{x-4}+2+\left|\sqrt{x-4}-2\right|}{1-\frac{4}{x}}\)

- Với \(x\ge8\Rightarrow\sqrt{x-4}-2\ge0\)

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

- Với \(4< x\le8\)

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