a: ĐKXĐ: x>4\(A=\dfrac{\sqrt{x-4+2\cdot\sqrt{x-4}\cdot2+4}+\sqrt{x-4-2\cdot\sqrt{x-4}\cdot2+4}}{\sqrt{\dfrac{x^2-8x+16}{x^2}}}\)
\(=\dfrac{\left|\sqrt{x-4}+2\right|+\left|\sqrt{x-4}-2\right|}{\dfrac{x-4}{x}}\)
\(=\left(\sqrt{x-4}+2+\left|\sqrt{x-4}-2\right|\right)\cdot\dfrac{x}{x-4}\)
TH1: x>=8
\(A=\left(\sqrt{x-4}+2+\sqrt{x-4}-2\right)\cdot\dfrac{x}{x-4}=\dfrac{2x}{\sqrt{x-4}}\)
TH2: 4<x<8
\(A=\left(\sqrt{x-4}+2+2-\sqrt{x-4}\right)\cdot\dfrac{x}{x-4}=\dfrac{4x}{x-4}\)
b: TH1: x>=8
Để A nguyên thì \(2x⋮\sqrt{x-4}\)
=>\(4x^2⋮x-4\)
\(\Leftrightarrow x-4\in\left\{1;-1;2;-2;4;-4;8;-8;16;-16\right\}\)
hay \(x\in\left\{8;12;20\right\}\)
TH2: 4<x<8
để A là số nguyên thì 4x chia hết cho x-4
=>\(x-4\in\left\{1;-1;2;-2;4;-4;8;-8;16;-16\right\}\)
hay \(x\in\left\{5;6\right\}\)
