a) \(\sqrt{\dfrac{x-2\sqrt{x}+1}{x+2\sqrt{x}+1}}=\sqrt{\dfrac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+1\right)^2}}=\left|\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\right|\)
Xét \(x\ge1\Rightarrow\left|\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\right|=\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\)
Xét \(0\le x< 1\Rightarrow\left|\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\right|=\dfrac{1-\sqrt{x}}{\sqrt{x}+1}\)
b) \(\dfrac{2}{x^2-y^2}\sqrt{\dfrac{3x^2+6xy+3y^2}{4}}=\dfrac{2}{\left(x-y\right)\left(x+y\right)}\sqrt{\dfrac{3\left(x+y\right)^2}{4}}\)
\(=\dfrac{2}{\left(x-y\right)\left(x+y\right)}\left|\dfrac{\sqrt{3}}{2}\left(x+y\right)\right|=\dfrac{2}{\left(x-y\right)\left(x+y\right)}.\dfrac{\sqrt{3}}{2}\left(x+y\right)=\dfrac{\sqrt{3}}{x-y}\)
c) \(\dfrac{x+\sqrt{7}}{x^2+2x\sqrt{7}+7}\left(x\ne-\sqrt{7}\right)=\dfrac{x+\sqrt{7}}{\left(x+\sqrt{7}\right)^2}=\dfrac{1}{x+\sqrt{7}}\)
d) \(\dfrac{x\sqrt{y}+y\sqrt{x}}{x+2\sqrt{xy}+y}=\dfrac{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)}{\left(\sqrt{x}+\sqrt{y}\right)^2}=\dfrac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\)
