\(J=\frac{(\sqrt{x}+1)^2-(\sqrt{x}-1)^2}{(\sqrt{x}-1)(\sqrt{x}+1)}:\frac{\sqrt{x}+1-(\sqrt{x}-1)}{\sqrt{x}+1}\)
\(=\frac{4\sqrt{x}}{(\sqrt{x}-1)(\sqrt{x+1})}:\frac{2}{\sqrt{x}+1}=\frac{4\sqrt{x}}{(\sqrt{x}+1)(\sqrt{x}-1)}.\frac{\sqrt{x}+1}{2}=\frac{2\sqrt{x}}{\sqrt{x}-1}\)
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\(K=\left[\frac{x}{\sqrt{x}(\sqrt{x}-1)}-\frac{1}{\sqrt{x}(\sqrt{x}-1)}\right]:\left[\frac{\sqrt{x}-1}{(\sqrt{x}-1)(\sqrt{x}+1)}+\frac{2}{(\sqrt{x}-1)(\sqrt{x}+1)}\right]\)
\(=\frac{x-1}{\sqrt{x}(\sqrt{x}-1)}:\frac{\sqrt{x}+1}{(\sqrt{x}-1)(\sqrt{x}+1)}=\frac{(\sqrt{x}-1)(\sqrt{x}+1)}{\sqrt{x}(\sqrt{x}-1)}:\frac{1}{\sqrt{x}-1}=\frac{x-1}{\sqrt{x}}\)
\(L=\left[\frac{\sqrt{a}-2}{(\sqrt{a}-1)(\sqrt{a}+1)}-\frac{\sqrt{a}+2}{(\sqrt{a}+1)^2}\right].\frac{\sqrt{a}+1}{\sqrt{a}}\)
\(=\frac{\sqrt{a}-2}{\sqrt{a}(\sqrt{a}-1)}-\frac{\sqrt{a}+2}{\sqrt{a}(\sqrt{a}+1)}\)
\(=\frac{(\sqrt{a}-2)(\sqrt{a}+1)-(\sqrt{a}+2)(\sqrt{a}-1)}{\sqrt{a}(\sqrt{a}-1)(\sqrt{a}+1)}=\frac{-2\sqrt{a}}{\sqrt{a}(a-1)}=\frac{2}{1-a}\)
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\(M=\frac{(\sqrt{x}+1)(\sqrt{x}+2)+2\sqrt{x}(\sqrt{x}-2)}{(\sqrt{x}-2)(\sqrt{x}+2)}-\frac{2+5\sqrt{x}}{x-4}\)
\(=\frac{x+3\sqrt{x}+2+2x-4\sqrt{x}}{(\sqrt{x}-2)(\sqrt{x}+2)}-\frac{2+5\sqrt{x}}{(\sqrt{x}-2)(\sqrt{x}+2)}=\frac{3x-\sqrt{x}+2-2-5\sqrt{x}}{(\sqrt{x}-2)(\sqrt{x}+2)}=\frac{3\sqrt{x}(\sqrt{x}-2)}{(\sqrt{x}-2)(\sqrt{x}+2)}=\frac{3\sqrt{x}}{\sqrt{x}+2}\)
\(N=\left[\frac{(\sqrt{x}+\sqrt{y})(x-\sqrt{xy}+y)}{\sqrt{x}+\sqrt{y}}-\sqrt{xy}\right].\left[\frac{\sqrt{x}+\sqrt{y}}{(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})}\right]^2\)
\(=(x-\sqrt{xy}+y-\sqrt{xy}).\frac{1}{(\sqrt{x}-\sqrt{y})^2}=(x+y-2\sqrt{xy}).\frac{1}{x+y-2\sqrt{xy}}=1\)
