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 - Page 661: 77



Work Step by Step

Multiplying by the conjugate of the numerator, the rationalized-numerator form of the given expression, $ \dfrac{\sqrt{a+h}-\sqrt{a}}{h} ,$ is \begin{array}{l}\require{cancel} \dfrac{\sqrt{a+h}-\sqrt{a}}{h}\cdot\dfrac{\sqrt{a+h}+\sqrt{a}}{\sqrt{a+h}+\sqrt{a}} \\\\= \dfrac{(\sqrt{a+h})^2-(\sqrt{a})^2}{h(\sqrt{a+h})+h(\sqrt{a})} \\\\= \dfrac{a+h-a}{h\sqrt{a+h}+h\sqrt{a}} \\\\= \dfrac{h}{h(\sqrt{a+h}+\sqrt{a})} \\\\= \dfrac{\cancel{h}}{\cancel{h}(\sqrt{a+h}+\sqrt{a})} \\\\= \dfrac{1}{\sqrt{a+h}+\sqrt{a}} .\end{array}
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