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
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Given,
Equation : 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
Simplifying the above equation :
⇒(7+35)×(3−5)−(7−35)×(3+5)(3+5)×(3−5)⇒(21−75+95−15)−(21+75−95−15)32−(5)2⇒21−75+95−15−21−75+95+159−5⇒454⇒5\Rightarrow \dfrac{(7 + 3\sqrt{5}) \times (3 - \sqrt{5}) - (7 - 3\sqrt{5}) \times (3 + \sqrt{5})}{(3 + \sqrt{5}) \times (3 - \sqrt{5})} \\[1em] \Rightarrow \dfrac{(21 - 7\sqrt{5} + 9\sqrt{5} - 15)-(21 + 7\sqrt{5} - 9\sqrt{5} - 15)}{3^2 - (\sqrt{5})^2} \\[1em] \Rightarrow \dfrac{21 - 7\sqrt{5} + 9\sqrt{5} - 15 - 21 - 7\sqrt{5} + 9\sqrt{5} + 15}{9 - 5} \\[1em] \Rightarrow \dfrac{4\sqrt{5}}{4} \\[1em] \Rightarrow \sqrt{5}⇒(3+5)×(3−5)(7+35)×(3−5)−(7−35)×(3+5)⇒32−(5)2(21−75+95−15)−(21+75−95−15)⇒9−521−75+95−15−21−75+95+15⇒445⇒5
Hence, 7+353+5−7−353−5=5\dfrac{7 + 3\sqrt{5}}{3 + \sqrt{5}}- \dfrac{7 - 3\sqrt{5}}{3 - \sqrt{5}} = \sqrt{5}3+57+35−3−57−35=5.
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