56. a) \(\left(\dfrac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}-\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\right):\dfrac{\sqrt{xy}}{x-y}\)
\(=\dfrac{\left(\sqrt{x}+\sqrt{y}\right)^2-\left(\sqrt{x}-\sqrt{y}\right)^2}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}.\dfrac{x-y}{\sqrt{xy}}=\dfrac{4\sqrt{xy}}{x-y}.\dfrac{x-y}{\sqrt{xy}}=4\)
b) \(\left(1+\dfrac{a+\sqrt{a}}{\sqrt{a}+1}\right).\left(1-\dfrac{a-\sqrt{a}}{\sqrt{a}-1}\right)\)
\(=\left(1+\dfrac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}+1}\right).\left(1-\dfrac{\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}-1}\right)\)
\(=\left(1+\sqrt{a}\right)\left(1-\sqrt{a}\right)=1-a\)
55.a) \(\dfrac{1}{\sqrt{x}-1}+\dfrac{1}{1+\sqrt{x}}+1=\dfrac{\sqrt{x}+1+\sqrt{x}-1+x-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{x+2\sqrt{x}-1}{x-1}\)
b) \(\dfrac{1}{\sqrt{x}+2}-\dfrac{2}{\sqrt{x}-2}-\dfrac{\sqrt{x}}{4-x}\)
\(=\dfrac{1}{\sqrt{x}+2}-\dfrac{2}{\sqrt{x}-2}+\dfrac{\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{\sqrt{x}-2-2\left(\sqrt{x}+2\right)+\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{-6}{x-4}\)
c) \(5\sqrt{\dfrac{x}{y}}-4\sqrt{\dfrac{y}{x}}+\sqrt{\dfrac{1}{xy}}=5\sqrt{\dfrac{x^2}{xy}}-4\sqrt{\dfrac{y^2}{xy}}+\sqrt{\dfrac{1}{xy}}\)
\(=5x\sqrt{\dfrac{1}{xy}}-4y\sqrt{\dfrac{1}{xy}}+\sqrt{\dfrac{1}{xy}}\left(x,y>0\right)\)
\(=\sqrt{\dfrac{1}{xy}}\left(5x-4y+1\right)\)
56:
a) Ta có: \(\left(\dfrac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}-\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\right):\dfrac{\sqrt{xy}}{x-y}\)
\(=\dfrac{x+2\sqrt{xy}+y-x+2\sqrt{xy}-y}{\left(\sqrt{x}-\sqrt{x}\right)\left(\sqrt{x}+\sqrt{y}\right)}\cdot\dfrac{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{xy}}\)
\(=\dfrac{4\sqrt{xy}}{\sqrt{xy}}=4\)
b) Ta có: \(\left(1+\dfrac{a+\sqrt{a}}{\sqrt{a}+1}\right)\left(1-\dfrac{a-\sqrt{a}}{\sqrt{a}-1}\right)\)
\(=\left(1+\sqrt{a}\right)\left(1-\sqrt{a}\right)\)
=1-a
