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$\frac{1}{3- \sqrt{7} } + \frac{23}{7+ \sqrt{3} } - \frac{2}{ \sqrt{7} + \sqrt{3} } = \\ = \frac{3+ \sqrt{7} }{(3- \sqrt{7})(3+\sqrt{7} ) } + \frac{23(7-\sqrt{3})}{(7+ \sqrt{3})(7- \sqrt{3}) } - \frac{2(\sqrt{7} - \sqrt{3})}{ (\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3}) } = \\ = \frac{3+ \sqrt{7} }{ 9-7 } + \frac{23(7-\sqrt{3})}{49-3 } - \frac{2(\sqrt{7} - \sqrt{3})}{ 7-3 }= \frac{3+ \sqrt{7} }{2 } + \frac{23(7-\sqrt{3})}{46 } - \frac{2(\sqrt{7} - \sqrt{3})}{4 } = \\$ $= \frac{3+ \sqrt{7} }{2 } + \frac{7-\sqrt{3}}{2} - \frac{\sqrt{7} - \sqrt{3}}{2 } = \frac{3+ \sqrt{7} +7-\sqrt{3}- \sqrt{7} + \sqrt{3} }{2 } = \frac{10}{2} =5$