\(a,A=\dfrac{1}{x+\sqrt{x}}-\dfrac{2\sqrt{x}}{x-1}+\dfrac{1}{x-\sqrt{x}}\\ =\dfrac{1}{\sqrt{x}\left(\sqrt{x}+1\right)}-\dfrac{2\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}+\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\\ =\dfrac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\dfrac{2x}{\sqrt{x}\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\dfrac{\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\\ =\dfrac{\sqrt{x}-1-2x+\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\\ =\dfrac{-2x+2\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\\ =\dfrac{-2\sqrt{x}\left(\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\\ =\dfrac{-2}{\sqrt{x}+1}\)
\(b,B=\left(\dfrac{a}{a-2\sqrt{a}}+\dfrac{a}{\sqrt{a}-2}\right):\dfrac{\sqrt{a}+1}{a-4\sqrt{a}+4}\\ =\left[\dfrac{a}{\sqrt{a}\left(\sqrt{a}-2\right)}+\dfrac{a}{\sqrt{a}-2}\right]\cdot\dfrac{a-4\sqrt{a}+4}{\sqrt{a}+1}\\ =\dfrac{\sqrt{a}+a}{\sqrt{a}-2}\cdot\dfrac{\left(\sqrt{a}-2\right)^2}{\sqrt{a}+1}\\ =\dfrac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}-2}\cdot\dfrac{\left(\sqrt{a}-2\right)^2}{\sqrt{a}+1}\\ =\sqrt{a}\left(\sqrt{a}-2\right)\\ =a-2\sqrt{a}\)
c: \(\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}+\dfrac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\)
\(=\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{xy}-1\right)-\left(\sqrt{xy}+\sqrt{x}\right)\left(\sqrt{xy}+1\right)+xy-1}{xy-1}\)
\(=\dfrac{x\sqrt{y}-\sqrt{x}+\sqrt{xy}-1-xy-\sqrt{xy}-x\sqrt{y}-\sqrt{x}+xy-1}{xy-1}\)
\(=\dfrac{-2\sqrt{x}-2}{xy-1}\)
\(1-\dfrac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}\)
\(=\dfrac{xy-1-\left(\sqrt{xy}+\sqrt{x}\right)\left(\sqrt{xy}+1\right)-\left(\sqrt{x}+1\right)\left(\sqrt{xy}-1\right)}{xy-1}\)
\(=\dfrac{xy-1-xy-\sqrt{xy}-x\sqrt{y}-\sqrt{x}-x\sqrt{y}+\sqrt{x}-\sqrt{xy}+1}{xy-1}\)
\(=\dfrac{-2\sqrt{xy}-2x\sqrt{y}}{xy-1}\)
\(C=\left(\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}+\dfrac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\right):\left(1-\dfrac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}\right)\)
\(=\dfrac{-2\sqrt{x}-2}{xy-1}:\dfrac{-2\sqrt{xy}-2x\sqrt{y}}{xy-1}\)
\(=\dfrac{-2\left(\sqrt{x}+1\right)}{-2\sqrt{xy}\left(\sqrt{x}+1\right)}=\dfrac{1}{\sqrt{xy}}\)
d: \(\dfrac{\sqrt{x}+2}{\sqrt{x}+1}-\dfrac{x-5}{x-\sqrt{x}-2}\)
\(=\dfrac{\sqrt{x}+2}{\sqrt{x}+1}-\dfrac{x-5}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)-x+5}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}=\dfrac{x-4-x+5}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
\(\dfrac{\sqrt{x}}{\sqrt{x}-2}-\dfrac{x-\sqrt{x}-1}{x-2\sqrt{x}}\)
\(=\dfrac{\sqrt{x}}{\sqrt{x}-2}-\dfrac{x-\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-2\right)}\)
\(=\dfrac{x-x+\sqrt{x}+1}{\left(\sqrt{x}-2\right)\cdot\sqrt{x}}=\dfrac{\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-2\right)}\)
\(D=\left(\dfrac{\sqrt{x}}{\sqrt{x}-2}-\dfrac{x-\sqrt{x}-1}{x-2\sqrt{x}}\right):\left(\dfrac{\sqrt{x}+2}{\sqrt{x}+1}-\dfrac{x-5}{x-\sqrt{x}-2}\right)\)
\(=\dfrac{\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-2\right)}:\dfrac{1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-2\right)}\cdot\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}{1}=\dfrac{\left(\sqrt{x}+1\right)^2}{\sqrt{x}}\)
b,B=(a−2aa+a−2a):a−4a+4a+1=[a(a−2)a+a−2a]⋅a+1a−4a+4=a−2a+a⋅a+1(a−2)2=a−2a(a+1)⋅a+1(a−2)2=a(a−2)=a−2a
