Elementary Algebra

Published by Cengage Learning
ISBN 10: 1285194055
ISBN 13: 978-1-28519-405-9

Chapter 9 - Roots and Radicals - 9.4 - Products and Quotients Involving Radicals - Problem Set 9.4: 59

Answer

$\sqrt{5}-\sqrt{3}$

Work Step by Step

Multiplying by the conjugate of the denominator and using $(a+b)(a-b)=a^2-b^2,$ the given expression, $ \dfrac{2}{\sqrt{5}+\sqrt{3}} ,$ simplifies to \begin{array}{l}\require{cancel} \dfrac{2}{\sqrt{5}+\sqrt{3}}\cdot\dfrac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}} \\\\= \dfrac{2(\sqrt{5}-\sqrt{3})}{(\sqrt{5})^2-(\sqrt{3})^2} \\\\= \dfrac{2(\sqrt{5}-\sqrt{3})}{5-3} \\\\= \dfrac{2(\sqrt{5}-\sqrt{3})}{2} \\\\= \dfrac{\cancel{2}(\sqrt{5}-\sqrt{3})}{\cancel{2}} \\\\= \sqrt{5}-\sqrt{3} .\end{array}
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