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: 79


The rationalized form of the expression $\frac{\sqrt{x+5}-\sqrt{x}}{5}$ is $\frac{1}{\left( \sqrt{x+5}+\sqrt{x} \right)}$.

Work Step by Step

Consider the provided expression, $\frac{\sqrt{x+5}-\sqrt{x}}{5}$ Multiply the numerator and denominator by $\sqrt{x+5}+\sqrt{x}$ so that the radicand in the numerator is a perfect square. Therefore, $\frac{\sqrt{x+5}-\sqrt{x}}{5}=\frac{\sqrt{x+5}-\sqrt{x}}{5}\cdot \frac{\sqrt{x+5}+\sqrt{x}}{\sqrt{x+5}+\sqrt{x}}$ Apply the difference of squares property in the numerator, $\left( \sqrt{x+5}+\sqrt{x} \right)\left( \sqrt{x+5}-\sqrt{x} \right)={{\left( \sqrt{x+5} \right)}^{2}}-{{\left( \sqrt{x} \right)}^{2}}$ Apply the power of a power property ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ in the expression ${{\left( \sqrt{x+5} \right)}^{2}},{{\left( \sqrt{x} \right)}^{2}}$. $\begin{align} & {{\left( \sqrt{x+5} \right)}^{2}}={{\left( x+5 \right)}^{2\cdot \frac{1}{2}}} \\ & ={{\left( x+5 \right)}^{1}} \\ & =x+5 \end{align}$ And, $\begin{align} & {{\left( \sqrt{x} \right)}^{2}}={{\left( x \right)}^{2\cdot \frac{1}{2}}} \\ & ={{\left( x \right)}^{1}} \\ & =x \end{align}$ Rewrite the original expression as: $\begin{align} & \frac{\sqrt{x+5}-\sqrt{x}}{5}=\frac{\sqrt{x+5}-\sqrt{x}}{5}\cdot \frac{\sqrt{x+5}+\sqrt{x}}{\sqrt{x+5}+\sqrt{x}} \\ & =\frac{{{\left( \sqrt{x+5} \right)}^{2}}-{{\left( \sqrt{x} \right)}^{2}}}{5\left( \sqrt{x+5}+\sqrt{x} \right)} \\ & =\frac{x+5-x}{5\left( \sqrt{x+5}+\sqrt{x} \right)} \\ & =\frac{5}{5\left( \sqrt{x+5}+\sqrt{x} \right)} \end{align}$ Further simplify, $\begin{align} & \frac{\sqrt{x+5}-\sqrt{x}}{5}=\frac{5}{5\left( \sqrt{x+5}+\sqrt{x} \right)} \\ & =\frac{1}{\left( \sqrt{x+5}+\sqrt{x} \right)} \end{align}$ Therefore, the rationalized form of the expression $\frac{\sqrt{x+5}-\sqrt{x}}{5}$ is $\frac{1}{\left( \sqrt{x+5}+\sqrt{x} \right)}$.
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