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: 78

Answer

$\dfrac{-1}{\sqrt{x-h}+\sqrt{x}}$

Work Step by Step

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