Simplify : 5+35−3+5−35+3\dfrac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} + \dfrac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}5−35+3+5+35−3
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Given,
Equation : 5+35−3+5−35+3\dfrac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} + \dfrac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}5−35+3+5+35−3
Simplifying the above equation :
⇒(5+3)2+(5−3)2(5−3)(5+3)⇒(5)2+(3)2+2×5×3+(5)2+(3)2−2×5×3(5)2−(3)2⇒5+3+215+5+3−2155−3⇒162⇒8\Rightarrow \dfrac{(\sqrt{5} + \sqrt{3})^2 + (\sqrt{5} - \sqrt{3})^2}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})} \\[1em] \Rightarrow \dfrac{(\sqrt{5})^2 + (\sqrt{3})^2 + 2 \times \sqrt{5} \times \sqrt{3} + (\sqrt{5})^2 + (\sqrt{3})^2 - 2 \times \sqrt{5} \times \sqrt{3}}{(\sqrt{5})^2 - (\sqrt{3})^2} \\[1em] \Rightarrow \dfrac{5 + 3 + 2\sqrt{15} + 5 + 3 - 2\sqrt{15}}{5 - 3} \\[1em] \Rightarrow \dfrac{16}{2} \\[1em] \Rightarrow 8⇒(5−3)(5+3)(5+3)2+(5−3)2⇒(5)2−(3)2(5)2+(3)2+2×5×3+(5)2+(3)2−2×5×3⇒5−35+3+215+5+3−215⇒216⇒8
Hence, 5+35−3+5−35+3\dfrac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}+\dfrac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}5−35+3+5+35−3 = 8.
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Simplify : 7+353+5−7−353−5\dfrac{7 + 3\sqrt{5}}{3 + \sqrt{5}} - \dfrac{7 - 3\sqrt{5}}{3 - \sqrt{5}}3+57+35−3−57−35
Show that : 1(3−8)+1(7−6)+1(5−2)−1(8−7)−1(6−5)=5\dfrac{1}{(3 - \sqrt{8})} + \dfrac{1}{(\sqrt{7} - \sqrt{6})}+ \dfrac{1}{(\sqrt{5} - 2)} - \dfrac{1}{(\sqrt{8} - \sqrt{7})} - \dfrac{1}{(\sqrt{6} - \sqrt{5})} = 5(3−8)1+(7−6)1+(5−2)1−(8−7)1−(6−5)1=5
