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 R - Elementary Algebra Review - R.6 Rational Expressions and Equations - R.6 Exercise Set - Page 980: 38


The simplified form of a complex rational expression $\frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}$ is $\frac{a+b}{a}$.

Work Step by Step

$\frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}$ The numerator and denominator of the complex rational expression are single. Divide the numerator by the denominator, $\frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}=\frac{a}{a-b}\div \frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}$ $\begin{align} & \frac{A}{B}\div \frac{C}{D}=\frac{A}{B}\cdot \frac{D}{C} \\ & =\frac{AD}{BC} \end{align}$ where $\frac{A}{B},\left( B\ne 0 \right)$, $\frac{C}{D},\left( D\ne 0 \right)$ are rational expressions with $\frac{C}{D}\ne 0$ The reciprocal of $\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}$ is $\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}}$ So, multiply the reciprocal of the divisor, $\frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}=\frac{a}{a-b}\cdot \frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}}$ Simplify the terms, $\begin{align} & \frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}=\frac{a\cdot \left( {{a}^{2}}-{{b}^{2}} \right)}{\left( a-b \right)\cdot \left( {{a}^{2}} \right)} \\ & =\frac{a}{{{a}^{2}}}\cdot \frac{\left( {{a}^{2}}-{{b}^{2}} \right)}{\left( a-b \right)} \end{align}$ Take the factor for the numerator term, $\begin{align} & \frac{\frac{a}{a-b}}{\frac{{{a}^{2}}}{{{a}^{2}}-{{b}^{2}}}}=\frac{a}{{{a}^{2}}}\cdot \frac{\left( a+b \right)\left( a-b \right)}{\left( a-b \right)} \\ & =\frac{1}{a}\cdot \left( a+b \right) \\ & =\frac{a+b}{a} \end{align}$
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