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Rationalize the denominator of \(\dfrac{1}{1 + \sqrt{3} + \sqrt{5}}\)

 Jul 9, 2020
 #1
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Multiply conjugates.

 

\(\quad \dfrac1{1 + \sqrt 3 + \sqrt 5}\\ = \dfrac1{1 + \sqrt 3 + \sqrt 5} \cdot \dfrac{1 + \sqrt 3 - \sqrt 5}{1 + \sqrt 3 - \sqrt 5}\\ = \dfrac{1 + \sqrt 3 - \sqrt 5}{(1 + \sqrt 3)^2 - \sqrt5^2}\\ = \dfrac{1 + \sqrt 3 - \sqrt 5}{4 + 2\sqrt 3 - 5}\\ = \dfrac{1 + \sqrt 3 - \sqrt 5}{2\sqrt 3 - 1}\\ = \dfrac{1 + \sqrt 3 - \sqrt 5}{2\sqrt 3 - 1}\cdot \dfrac{2\sqrt 3 + 1}{2\sqrt 3 + 1}\\ = \dfrac{(1 + \sqrt 3 - \sqrt 5)(2 \sqrt 3 + 1)}{(2\sqrt 3)^2 - 1^2}\\ = \dfrac{1 + 2\sqrt 3 + 6 + \sqrt 3 - 2\sqrt{15} - \sqrt 5}{11}\\ = \dfrac{7 + 3 \sqrt 3 - \sqrt 5 - 2\sqrt {15}}{11}\)

 Jul 9, 2020

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