Simplify :
717−23−317+23\dfrac{7}{\sqrt{17} - 2\sqrt{3}} - \dfrac{3}{\sqrt{17} + 2\sqrt{3}}17−237−17+233
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717−23−317+23=7×(17+23)−3×(17−23)(17−23)×(17+23)=(717+143)−(317−63)(17)2−(23)2=717+143−317+6317−12=417+2035\dfrac{7}{\sqrt{17} - 2\sqrt{3}} - \dfrac{3}{\sqrt{17} + 2\sqrt{3}}\\[1em] = \dfrac{7 \times (\sqrt{17} + 2\sqrt{3}) - 3 \times (\sqrt{17} - 2\sqrt{3})}{(\sqrt{17} - 2\sqrt{3}) \times (\sqrt{17} + 2\sqrt{3})} \\[1em] = \dfrac{(7\sqrt{17} + 14\sqrt{3}) - (3\sqrt{17} - 6\sqrt{3})}{(\sqrt{17})^2 - (2\sqrt{3})^2}\\[1em] = \dfrac{7\sqrt{17} + 14\sqrt{3} - 3\sqrt{17} + 6\sqrt{3}}{17 - 12}\\[1em] = \dfrac{4\sqrt{17} + 20\sqrt{3}}{5}17−237−17+233=(17−23)×(17+23)7×(17+23)−3×(17−23)=(17)2−(23)2(717+143)−(317−63)=17−12717+143−317+63=5417+203
Hence, 717−23−317+23=417+2035\dfrac{7}{\sqrt{17} - 2\sqrt{3}} - \dfrac{3}{\sqrt{17} + 2\sqrt{3}} = \dfrac{4\sqrt{17} + 20\sqrt{3}}{5}17−237−17+233=5417+203.
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35−3+25+3\dfrac{3}{5 - \sqrt3} + \dfrac{2}{5 + \sqrt3}5−33+5+32
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
Find the value of m and n: if:
3+23−2=m+n2\dfrac{3 + \sqrt2}{3 - \sqrt2} = m + n \sqrt23−23+2=m+n2
5+237+43=m+n3\dfrac{5 + 2\sqrt3}{7 + 4\sqrt3} = m + n\sqrt37+435+23=m+n3
