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\(\cfrac{\sqrt5}{\sqrt3+\sqrt2}-\cfrac{3\sqrt3}{\sqrt2 +\sqrt5}+ \cfrac{2\sqrt2}{\sqrt3+ \sqrt5}\)
\(=\cfrac{\sqrt5}{\sqrt3-\sqrt2}-\cfrac{3\sqrt3}{\sqrt5+\sqrt2}+ \cfrac{2\sqrt2}{\sqrt5+\sqrt3}\)
\(=\cfrac{\sqrt5(\sqrt3-\sqrt2)}{(\sqrt3+\sqrt2)(\sqrt3-\sqrt2)}\)

\(-\cfrac{3\sqrt3(\sqrt5-\sqrt2)}{(\sqrt5+\sqrt2)(\sqrt5-\sqrt2)}\)

\(+ \cfrac{2\sqrt2(\sqrt5-\sqrt3)}{(\sqrt5+\sqrt3)(\sqrt5-\sqrt3)}\)

\(=\cfrac{\sqrt5(\sqrt3-\sqrt2)}{(\sqrt3)^2-(\sqrt2)^2} -\cfrac{3\sqrt3(\sqrt5-\sqrt2)}{(\sqrt5)^2-(\sqrt2)^2}\)

\(+ \cfrac{2\sqrt2(\sqrt5-\sqrt3)}{(\sqrt5)^2-(\sqrt3)^2}\)

\(=\cfrac{\sqrt5(\sqrt3-\sqrt2)}{3-2} -\cfrac{3\sqrt3(\sqrt5-\sqrt2)}{5-2}\)

\(+ \cfrac{2\sqrt2(\sqrt5-\sqrt3)}{5-3}\)

\(=\cfrac{\sqrt5(\sqrt3-\sqrt2)}{1} -\cfrac{\cancel3\sqrt3(\sqrt5-\sqrt2)}{\cancel3}\)

\(+ \cfrac{\cancel2\sqrt2(\sqrt5-\sqrt3)}{\cancel2}\)

\(=\cancel{\sqrt{15}}-\cancel{\sqrt{10}}-\cancel{\sqrt{15}}+\cancel{\sqrt6}\)

\(+\cancel{\sqrt{10}}-\cancel{\sqrt6}\)

\(=0\)