rodneywalker
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# What is the simplest form of the expression? sqrt 2 - sqrt 6 / sqrt 2 + sqrt 6

$\displaystyle \\ \frac{ \sqrt{2} -\sqrt{6}}{\sqrt{2} +\sqrt{6}} = \frac{ (\sqrt{2} -\sqrt{6})(\sqrt{2} -\sqrt{6})}{(\sqrt{2} +\sqrt{6})(\sqrt{2} -\sqrt{6})} = \\ \\ \\ = \frac{ (\sqrt{2} -\sqrt{6})^2}{(\sqrt{2})^2 -(\sqrt{6})^2} = \frac{ 2-2\sqrt{2}\sqrt{6} + 6}{2 -6} = \\ \\ \\ = \frac{ 8-2\sqrt{12} }{-4} = \frac{ 8-2\sqrt{4\times 3} }{-4} = \frac{ 8-4\sqrt{3} }{-4} = \boxed{-2+\sqrt{3}}$