Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 10 - Exponents and Radicals - 10.5 Expressions Containing Several Radical Terms - 10.5 Exercise Set: 75

Answer

$\dfrac{x-y}{x+2\sqrt{xy}+y}$

Work Step by Step

Multiplying by the conjugate of the numerator, the rationalized-numerator form of the given expression, $ \dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}} ,$ is \begin{array}{l}\require{cancel} \dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\cdot\dfrac{\sqrt{x}+\sqrt{y}}{\sqrt{x}+\sqrt{y}} \\\\= \dfrac{(\sqrt{x})^2-(\sqrt{y})^2}{\sqrt{x}(\sqrt{x})+\sqrt{x}(\sqrt{y})+\sqrt{y}(\sqrt{x})+\sqrt{y}(\sqrt{y})} \\\\= \dfrac{x-y}{\sqrt{x(x)}+\sqrt{x(y)}+\sqrt{y(x)}+\sqrt{y(y)}} \\\\= \dfrac{x-y}{\sqrt{(x)^2}+\sqrt{xy}+\sqrt{xy}+\sqrt{(y)^2}} \\\\= \dfrac{x-y}{x+2\sqrt{xy}+y} .\end{array}
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