College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter P, Prerequisites - Section P.7 - Rational Expressions - P.7 Exercises: 74



Work Step by Step

First we form a common denominator to combine the top fractions. Then we multiply by $\sqrt{x}+\sqrt{x+h}$ and use the fact that $(a-b)(a+b)=a^b-b^2$: $\displaystyle \frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}=\frac{\frac{\sqrt{x}}{\sqrt{x}\sqrt{x+h}}-\frac{\sqrt{x+h}}{\sqrt{x+h}\sqrt{x}}}{h}=\frac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x}\sqrt{x+h}}\cdot\frac{\sqrt{x}+\sqrt{x+h}}{\sqrt{x}+\sqrt{x+h}}=\frac{x-(x+h)}{h\sqrt{x}\sqrt{x+h}(\sqrt{x}+\sqrt{x+h})}=-\frac{1}{\sqrt{x}\sqrt{x+h}(\sqrt{x}+\sqrt{x+h})}$
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