Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter P - Section P.6 - Rational Expressions - Exercise Set - Page 85: 78


The simplified form of the expression $\frac{\frac{1}{\sqrt{x+3}}-\frac{1}{\sqrt{x}}}{3}$ is $\frac{\sqrt{x}-\sqrt{x+3}}{3\sqrt{x\left( x+3 \right)}}$.

Work Step by Step

Consider the provided expression, $\frac{\frac{1}{\sqrt{x+3}}-\frac{1}{\sqrt{x}}}{3}$ Multiply the numerator and denominator by $\sqrt{x}\sqrt{x+3}$. Therefore, $\frac{\frac{1}{\sqrt{x+3}}-\frac{1}{\sqrt{x}}}{3}=\frac{\frac{1}{\sqrt{x+3}}-\frac{1}{\sqrt{x}}}{3}\cdot \frac{\sqrt{x}\sqrt{x+3}}{\sqrt{x}\sqrt{x+3}}$ Apply the distributive property in the numerator: $a\left( b+c \right)=ab+ac$ Therefore, $$ $\begin{align} & \frac{\frac{1}{\sqrt{x+3}}-\frac{1}{\sqrt{x}}}{3}=\frac{\frac{1}{\sqrt{x+3}}\sqrt{x}\sqrt{x+3}-\frac{1}{\sqrt{x}}\sqrt{x}\sqrt{x+3}}{3\sqrt{x}\sqrt{x+3}} \\ & =\frac{\sqrt{x}-\sqrt{x+3}}{3\sqrt{x}\sqrt{x+3}} \\ & =\frac{\sqrt{x}+\sqrt{x+3}}{3\sqrt{x\left( x+3 \right)}} \end{align}$ $$ Therefore, the simplified form of the expression $\frac{\frac{1}{\sqrt{x+3}}-\frac{1}{\sqrt{x}}}{3}$ is $\frac{\sqrt{x}-\sqrt{x+3}}{3\sqrt{x\left( x+3 \right)}}$.
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