Simplify :
5−105+10−5+105−10\dfrac{5 - \sqrt{10}}{5 + \sqrt{10}} - \dfrac{5 + \sqrt{10}}{5 - \sqrt{10}}5+105−10−5−105+10
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5−105+10−5+105−10=(5−10)×(5−10)−(5+10)×(5+10)(5+10)×(5−10)=(5−10)2−(5+10)2(5)2−(10)2=(25+10−1010)−(25+10+1010)25−10=(35−1010)−(35+1010)15=35−1010−35−101015=−201015=−4310\dfrac{5 - \sqrt{10}}{5 + \sqrt{10}} - \dfrac{5 + \sqrt{10}}{5 - \sqrt{10}}\\[1em] = \dfrac{(5 - \sqrt{10}) \times (5 - \sqrt{10}) - (5 + \sqrt{10}) \times (5 + \sqrt{10})}{(5 + \sqrt{10}) \times (5 - \sqrt{10})}\\[1em] = \dfrac{(5 - \sqrt{10})^2 - (5 + \sqrt{10})^2}{(5)^2 - (\sqrt{10})^2}\\[1em] = \dfrac{(25 + 10 - 10\sqrt{10}) - (25 + 10 + 10\sqrt{10})}{25 - 10}\\[1em] = \dfrac{(35 - 10\sqrt{10}) - (35 + 10\sqrt{10})}{15}\\[1em] = \dfrac{35 - 10\sqrt{10} - 35 - 10\sqrt{10}}{15}\\[1em] = \dfrac{- 20\sqrt{10}}{15}\\[1em] = \dfrac{- 4}{3}\sqrt{10}5+105−10−5−105+10=(5+10)×(5−10)(5−10)×(5−10)−(5+10)×(5+10)=(5)2−(10)2(5−10)2−(5+10)2=25−10(25+10−1010)−(25+10+1010)=15(35−1010)−(35+1010)=1535−1010−35−1010=15−2010=3−410
Hence, 5−105+10−5+105−10=−4310\dfrac{5 - \sqrt{10}}{5 + \sqrt{10}} - \dfrac{5 + \sqrt{10}}{5 - \sqrt{10}} = \dfrac{- 4}{3}\sqrt{10}5+105−10−5−105+10=3−410.
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