Simplify :
35−3+25+3\dfrac{3}{5 - \sqrt3} + \dfrac{2}{5 + \sqrt3}5−33+5+32
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35−3+25+3=3×(5+3)+2×(5−3)(5−3)×(5+3)=(15+33)+(10−23)(5)2−(3)2=15+33+10−2322=25+322\dfrac{3}{5 - \sqrt3} + \dfrac{2}{5 + \sqrt3}\\[1em] = \dfrac{3 \times (5 + \sqrt3) + 2 \times (5 - \sqrt3)}{(5 - \sqrt3) \times (5 + \sqrt3)}\\[1em] = \dfrac{(15 + 3\sqrt3) + (10 - 2\sqrt3)}{(5)^2 - (\sqrt3)^2}\\[1em] = \dfrac{15 + 3\sqrt3 + 10 - 2\sqrt3}{22}\\[1em] = \dfrac{25 + \sqrt3}{22}5−33+5+32=(5−3)×(5+3)3×(5+3)+2×(5−3)=(5)2−(3)2(15+33)+(10−23)=2215+33+10−23=2225+3
Hence, 35−3+25+3=25+322\dfrac{3}{5 - \sqrt3} + \dfrac{2}{5 + \sqrt3} = \dfrac{25 + \sqrt3}{22}5−33+5+32=2225+3.
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Rationalise the denominator and simplify :
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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
717−23−317+23\dfrac{7}{\sqrt{17} - 2\sqrt{3}} - \dfrac{3}{\sqrt{17} + 2\sqrt{3}}17−237−17+233
