Question

Với x>=0 rút gọn biểu thức \(\sqrt{\frac{x+4-4\sqrt{x}}{x+4+4\sqrt{x}}}\)

Answer 1

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

TH 1 : \(x\ge4\Rightarrow\sqrt{x}\ge2\Rightarrow\sqrt{x}-2\ge0\)

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

TH 2 : \(0\le x< 4\Rightarrow\sqrt{x}< 2\Rightarrow\sqrt{x}-2< 0\)

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

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