Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 1 - Section 1.4 - Rational Expressions - 1.4 Exercises: 95

Answer

$\sqrt{x^{2}+1}-x=\dfrac{1}{\sqrt{x^{2}+1}+x}$

Work Step by Step

$\sqrt{x^{2}+1}-x$ Multiply this expression by $\dfrac{\sqrt{x^{2}+1}+x}{\sqrt{x^{2}+1}+x}$ and simplify: $\sqrt{x^{2}+1}-x=(\sqrt{x^{2}+1}-x)\cdot\dfrac{\sqrt{x^{2}+1}+x}{\sqrt{x^{2}+1}+x}=...$ $...=\dfrac{(\sqrt{x^{2}+1})^{2}-x^{2}}{\sqrt{x^{2}+1}+x}=\dfrac{x^{2}+1-x^{2}}{\sqrt{x^{2}+1}+x}=\dfrac{1}{\sqrt{x^{2}+1}+x}$
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