# Rational expressions: multiplying and dividing rational expressions

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<para>This module is from<link document="col10614">Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr.</para><para>A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.</para><para>The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.</para><para>Objectives of this module: be able to multiply and divide rational expressions.</para>

## Overview

• Multiplication Of Rational Expressions
• Division Of Rational Expressions

## Multiplication of rational expressions

Rational expressions are multiplied together in much the same way that arithmetic fractions are multiplied together. To multiply rational numbers, we do the following:

Method for Multiplying Rational Numbers
1. Reduce each fraction to lowest terms.
2. Multiply the numerators together.
3. Multiply the denominators together.

Rational expressions are multiplied together using exactly the same three steps. Since rational expressions tend to be longer than arithmetic fractions, we can simplify the multiplication process by adding one more step.

Method for Multiplying Rational Expressions
1. Factor all numerators and denominators.
2. Reduce to lowest terms first by dividing out all common factors. (It is perfectly legitimate to cancel the numerator of one fraction with the denominator of another.)
3. Multiply numerators together.
4. Multiply denominators. It is often convenient, but not necessary, to leave denominators in factored form.

## Sample set a

Perform the following multiplications.

$\frac{3}{4}·\frac{1}{2}=\frac{3·1}{4·2}=\frac{3}{8}$

$\frac{8}{9}·\frac{1}{6}=\frac{\stackrel{4}{\overline{)8}}}{9}·\frac{1}{\underset{3}{\overline{)6}}}=\frac{4·1}{9·3}=\frac{4}{27}$

$\frac{3x}{5y}·\frac{7}{12y}=\frac{\stackrel{1}{\overline{)3}}x}{5y}·\frac{7}{\underset{4}{\overline{)12}}y}=\frac{x·7}{5y·4y}=\frac{7x}{20{y}^{2}}$

$\begin{array}{lll}\frac{x+4}{x-2}·\frac{x+7}{x+4}\hfill & \hfill & \text{Divide out the common factor\hspace{0.17em}}x+4.\hfill \\ \frac{\overline{)x+4}}{x-2}·\frac{x+7}{\overline{)x+4}}\hfill & \hfill & \text{Multiply numerators and denominators together}\text{.}\hfill \\ \frac{x+7}{x-2}\hfill & \hfill & \hfill \end{array}$

$\begin{array}{l}\begin{array}{lll}\frac{{x}^{2}+x-6}{{x}^{2}-4x+3}·\frac{{x}^{2}-2x-3}{{x}^{2}+4x-12}.\hfill & \hfill & \text{Factor}\text{.}\hfill \\ \frac{\left(x+3\right)\left(x-2\right)}{\left(x-3\right)\left(x-1\right)}·\frac{\left(x-3\right)\left(x+1\right)}{\left(x+6\right)\left(x-2\right)}\hfill & \hfill & \text{Divide out the common factors\hspace{0.17em}}x-2\text{\hspace{0.17em}and\hspace{0.17em}}x-3.\hfill \\ \frac{\left(x+3\right)\overline{)\left(x-2\right)}}{\overline{)\left(x-3\right)}\left(x-1\right)}·\frac{\overline{)\left(x-3\right)}\left(x+1\right)}{\left(x+6\right)\overline{)\left(x-2\right)}}\hfill & \hfill & \text{Multiply}.\hfill \end{array}\\ \begin{array}{lllllllll}\frac{\left(x+3\right)\left(x+1\right)}{\left(x-1\right)\left(x+6\right)}\hfill & \hfill & \text{or}\hfill & \hfill & \frac{{x}^{2}+4x+3}{\left(x-1\right)\left(x+6\right)}\hfill & \hfill & \text{or}\hfill & \hfill & \frac{{x}^{2}+4x+3}{{x}^{2}+5x-6}\hfill \end{array}\\ \text{\hspace{0.17em}}\\ \text{Each of these three forms is an acceptable form of the same answer}.\end{array}$

$\begin{array}{l}\begin{array}{lll}\frac{2x+6}{8x-16}·\frac{{x}^{2}-4}{{x}^{2}-x-12}.\hfill & \hfill & \text{Factor}\text{.}\hfill \\ \frac{2\left(x+3\right)}{8\left(x-2\right)}·\frac{\left(x+2\right)\left(x-2\right)}{\left(x-4\right)\left(x+3\right)}\hfill & \hfill & \text{Divide out the common factors 2,\hspace{0.17em}}x+3\text{\hspace{0.17em}and\hspace{0.17em}}x-2.\hfill \\ \frac{\stackrel{1}{\overline{)2}}\overline{)\left(x+3\right)}}{\underset{4}{\overline{)8}}\overline{)\left(x-2\right)}}·\frac{\left(x+2\right)\overline{)\left(x-2\right)}}{\overline{)\left(x+3\right)}\left(x-4\right)}\hfill & \hfill & \text{Multiply}.\hfill \end{array}\\ \begin{array}{ccc}\frac{x+2}{4\left(x-4\right)}& \text{or}& \frac{x+2}{4x-16}\end{array}\\ \text{\hspace{0.17em}}\\ \text{Both these forms are acceptable forms of the same answer}.\end{array}$

$\begin{array}{lll}3{x}^{2}·\frac{x+7}{x-5}.\hfill & \hfill & \text{Rewrite\hspace{0.17em}}3{x}^{2}\text{\hspace{0.17em}as\hspace{0.17em}}\frac{3{x}^{2}}{1}.\hfill \\ \frac{3{x}^{2}}{1}·\frac{x+7}{x-5}\hfill & \hfill & \text{Multiply}.\hfill \\ \frac{3{x}^{2}\left(x+7\right)}{x-5}\hfill & \hfill & \hfill \end{array}$

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