# 8.8 Complex rational expressions

 Page 1 / 1
<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 distinguish between simple and complex fractions, be able to simplify complex fractions using the combine-divide and the LCD-multiply-divide method.</para>

## Overview

• Simple And Complex Fractions
• The Combine-Divide Method
• The LCD-Multiply-Divide Method

## Simple fraction

In Section [link] we saw that a simple fraction was a fraction of the form $\frac{P}{Q},$ where $P$ and $Q$ are polynomials and $Q\ne 0$ .

## Complex fraction

A complex fraction is a fraction in which the numerator or denominator, or both, is a fraction. The fractions $\begin{array}{lllll}\frac{\frac{8}{15}}{\frac{2}{3}}\hfill & \hfill & \text{and}\hfill & \hfill & \frac{1-\frac{1}{x}}{1-\frac{1}{{x}^{2}}}\hfill \end{array}$ are examples of complex fractions, or more generally, complex rational expressions.

There are two methods for simplifying complex rational expressions: the combine-divide method and the LCD-multiply-divide method.

## The combine-divide method

1. If necessary, combine the terms of the numerator together.
2. If necessary, combine the terms of the denominator together.
3. Divide the numerator by the denominator.

## Sample set a

Simplify each complex rational expression.

$\begin{array}{l}\frac{\frac{{x}^{3}}{8}}{\frac{{x}^{5}}{12}}\\ \\ \text{Steps\hspace{0.17em}1\hspace{0.17em}and\hspace{0.17em}2\hspace{0.17em}are\hspace{0.17em}not\hspace{0.17em}necessary\hspace{0.17em}so\hspace{0.17em}we\hspace{0.17em}proceed\hspace{0.17em}with\hspace{0.17em}step\hspace{0.17em}3}\text{.}\\ \\ \frac{\frac{{x}^{3}}{8}}{\frac{{x}^{5}}{12}}=\frac{{x}^{3}}{8}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{12}{{x}^{5}}=\frac{\overline{){x}^{3}}}{\underset{2}{\overline{)8}}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{\stackrel{3}{\overline{)12}}}{{x}^{\overline{)5}2}}=\frac{3}{2{x}^{2}}\end{array}$

$\begin{array}{l}\frac{1-\frac{1}{x}}{1-\frac{1}{{x}^{2}}}\\ \\ \text{Step\hspace{0.17em}1:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Combine\hspace{0.17em}the\hspace{0.17em}terms\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}numerator:\hspace{0.17em}LCD}=x\text{.}\\ 1-\frac{1}{x}=\frac{x}{x}-\frac{1}{x}=\frac{x-1}{x}\\ \\ \text{Step\hspace{0.17em}2:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Combine\hspace{0.17em}the\hspace{0.17em}terms\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}denominator:\hspace{0.17em}LCD}={x}^{2}\\ \\ 1-\frac{1}{{x}^{2}}=\frac{{x}^{2}}{{x}^{2}}-\frac{1}{{x}^{2}}=\frac{{x}^{2}-1}{{x}^{2}}\\ \\ \text{Step\hspace{0.17em}3:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Divide\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}by\hspace{0.17em}the\hspace{0.17em}denominator}\text{.}\\ \\ \begin{array}{lll}\frac{\frac{x-1}{x}}{\frac{{x}^{2}-1}{{x}^{2}}}\hfill & =\hfill & \frac{x-1}{x}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{{x}^{2}}{{x}^{2}-1}\hfill \\ \hfill & =\hfill & \frac{\overline{)x-1}}{\overline{)x}}\frac{{x}^{\overline{)2}}}{\left(x+1\right)\overline{)\left(x-1\right)}}\hfill \\ \hfill & =\hfill & \frac{x}{x+1}\hfill \end{array}\\ \\ \text{Thus,}\\ \\ \frac{1-\frac{1}{x}}{1-\frac{1}{{x}^{2}}}=\frac{x}{x+1}\end{array}$

$\begin{array}{l}\frac{2-\frac{13}{m}-\frac{7}{{m}^{2}}}{2+\frac{3}{m}+\frac{1}{{m}^{2}}}\\ \\ \text{Step\hspace{0.17em}1:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Combine\hspace{0.17em}the\hspace{0.17em}terms\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}numerator:\hspace{0.17em}LCD}={m}^{2}\text{.}\\ \\ 2-\frac{13}{m}-\frac{7}{{m}^{2}}=\frac{2{m}^{2}}{{m}^{2}}-\frac{13m}{{m}^{2}}-\frac{7}{{m}^{2}}=\frac{2{m}^{2}-13m-7}{{m}^{2}}\\ \\ \text{Step\hspace{0.17em}2:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Combine\hspace{0.17em}the\hspace{0.17em}terms\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}denominator:\hspace{0.17em}LCD}={m}^{2}\\ \\ 2+\frac{3}{m}+\frac{1}{{m}^{2}}=\frac{2{m}^{2}}{{m}^{2}}+\frac{3m}{{m}^{2}}+\frac{1}{{m}^{2}}=\frac{2{m}^{2}+3m+1}{{m}^{2}}\\ \\ \text{Step\hspace{0.17em}3:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Divide\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}by\hspace{0.17em}the\hspace{0.17em}denominator}\text{.}\\ \\ \begin{array}{lll}\frac{\frac{2{m}^{2}-13m-7}{{m}^{2}}}{\frac{2{m}^{2}+3m-1}{{m}^{2}}}\hfill & =\hfill & \frac{2{m}^{2}-13m-7}{{m}^{2}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{{m}^{2}}{2{m}^{2}+3m+1}\hfill \\ \hfill & =\hfill & \frac{\overline{)\left(2m+1\right)}\left(m-7\right)}{\overline{){m}^{2}}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{\overline{){m}^{2}}}{\overline{)\left(2m+1\right)}\left(m+1\right)}\hfill \\ \hfill & =\hfill & \frac{m-7}{m+1}\hfill \end{array}\\ \\ \text{Thus,}\\ \\ \frac{2-\frac{13}{m}-\frac{7}{{m}^{2}}}{2+\frac{3}{m}+\frac{1}{{m}^{2}}}=\frac{m-7}{m+1}\end{array}$

## Practice set a

Use the combine-divide method to simplify each expression.

$\frac{\frac{27{x}^{2}}{6}}{\frac{15{x}^{3}}{8}}$

$\frac{12}{5x}$

$\frac{3-\frac{1}{x}}{3+\frac{1}{x}}$

$\frac{3x-1}{3x+1}$

$\frac{1+\frac{x}{y}}{x-\frac{{y}^{2}}{x}}$

$\frac{x}{y\left(x-y\right)}$

$\frac{m-3+\frac{2}{m}}{m-4+\frac{3}{m}}$

$\frac{m-2}{m-3}$

$\frac{1+\frac{1}{x-1}}{1-\frac{1}{x-1}}$

$\frac{x}{x-2}$

## The lcd-multiply-divide method

1. Find the LCD of all the terms.
2. Multiply the numerator and denominator by the LCD.
3. Reduce if necessary.

## Sample set b

Simplify each complex fraction.

$\begin{array}{l}\frac{1-\frac{4}{{a}^{2}}}{1+\frac{2}{a}}\\ \\ \text{Step\hspace{0.17em}1:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}The\hspace{0.17em}LCD}={a}^{2}\text{.}\\ \text{Step\hspace{0.17em}2:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Multiply\hspace{0.17em}both\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}and\hspace{0.17em}denominator\hspace{0.17em}by\hspace{0.17em}}{a}^{2}\text{.}\\ \\ \begin{array}{lll}\frac{{a}^{2}\left(1-\frac{4}{{a}^{2}}\right)}{{a}^{2}\left(1+\frac{2}{a}\right)}\hfill & =\hfill & \frac{{a}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}1-{a}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{4}{{a}^{2}}}{{a}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}1+{a}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{2}{a}}\hfill \\ \hfill & =\hfill & \frac{{a}^{2}-4}{{a}^{2}+2a}\hfill \end{array}\\ \\ \text{Step\hspace{0.17em}3:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Reduce}\text{.}\\ \\ \begin{array}{lll}\frac{{a}^{2}-4}{{a}^{2}+2a}\hfill & =\hfill & \frac{\overline{)\left(a+2\right)}\left(a-2\right)}{a\overline{)\left(a+2\right)}}\hfill \\ \hfill & =\hfill & \frac{a-2}{a}\hfill \end{array}\\ \\ \text{Thus,}\\ \\ \frac{1-\frac{4}{{a}^{2}}}{1+\frac{2}{a}}=\frac{a-2}{a}\end{array}$

$\begin{array}{l}\begin{array}{l}\frac{1-\frac{5}{x}-\frac{6}{{x}^{2}}}{1+\frac{6}{x}+\frac{5}{{x}^{2}}}\hfill \\ \hfill \\ \text{Step}\text{\hspace{0.17em}}1:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{The}\text{\hspace{0.17em}}\text{LCD}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}{x}^{2}.\hfill \\ \text{Step}\text{\hspace{0.17em}}2:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Multiply}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{numerator}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{denominator}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}{x}^{2}.\hfill \\ \hfill \\ \begin{array}{lll}\frac{{x}^{2}\left(1-\frac{5}{x}-\frac{6}{{x}^{2}}\right)}{{x}^{2}\left(1+\frac{6}{x}+\frac{5}{{x}^{2}}\right)}\hfill & =\hfill & \frac{{x}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}1-{x}^{\overline{)2}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{5}{\overline{)x}}-\overline{){x}^{2}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{6}{\overline{){x}^{2}}}}{{x}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}1+{x}^{\overline{)2}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{6}{\overline{)x}}+\overline{){x}^{2}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{5}{\overline{){x}^{2}}}}\hfill \\ \hfill & =\hfill & \frac{{x}^{2}-5x-6}{{x}^{2}+6x+5}\hfill \end{array}\hfill \end{array}\\ \text{Step}\text{\hspace{0.17em}}3:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Reduce}\text{.}\\ \begin{array}{lll}\frac{{x}^{2}-5x-6}{{x}^{2}+6x+5}\hfill & =\hfill & \frac{\left(x-6\right)\left(x+1\right)}{\left(x+5\right)\left(x+1\right)}\hfill \\ \hfill & =\hfill & \frac{x-6}{x+5}\hfill \end{array}\\ \text{Thus,}\\ \frac{1-\frac{5}{x}-\frac{6}{{x}^{2}}}{1+\frac{6}{x}+\frac{5}{{x}^{2}}}=\frac{x-6}{x+5}\end{array}$

## Practice set b

The following problems are the same problems as the problems in Practice Set A. Simplify these expressions using the LCD-multiply-divide method. Compare the answers to the answers produced in Practice Set A.

$\frac{\frac{27{x}^{2}}{6}}{\frac{15{x}^{3}}{8}}$

$\frac{12}{5x}$

$\frac{3-\frac{1}{x}}{3+\frac{1}{x}}$

$\frac{3x-1}{3x+1}$

$\frac{1+\frac{x}{y}}{x-\frac{{y}^{2}}{x}}$

$\frac{x}{y\left(x-y\right)}$

$\frac{m-3+\frac{2}{m}}{m-4+\frac{3}{m}}$

$\frac{m-2}{m-3}$

$\frac{1+\frac{1}{x-1}}{1-\frac{1}{x-1}}$

$\frac{x}{x-2}$

## Exercises

For the following problems, simplify each complex rational expression.

$\frac{1+\frac{1}{4}}{1-\frac{1}{4}}$

$\frac{5}{3}$

$\frac{1-\frac{1}{3}}{1+\frac{1}{3}}$

$\frac{1-\frac{1}{y}}{1+\frac{1}{y}}$

$\frac{y-1}{y+1}$

$\frac{a+\frac{1}{x}}{a-\frac{1}{x}}$

$\frac{\frac{a}{b}+\frac{c}{b}}{\frac{a}{b}-\frac{c}{b}}$

$\frac{a+c}{a-c}$

$\frac{\frac{5}{m}+\frac{4}{m}}{\frac{5}{m}-\frac{4}{m}}$

$\frac{3+\frac{1}{x}}{\frac{3x+1}{{x}^{2}}}$

$x$

$\frac{1+\frac{x}{x+y}}{1-\frac{x}{x+y}}$

$\frac{2+\frac{5}{a+1}}{2-\frac{5}{a+1}}$

$\frac{2a+7}{2a-3}$

$\frac{1-\frac{1}{a-1}}{1+\frac{1}{a-1}}$

$\frac{4-\frac{1}{{m}^{2}}}{2+\frac{1}{m}}$

$\frac{2m-1}{m}$

$\frac{9-\frac{1}{{x}^{2}}}{3-\frac{1}{x}}$

$\frac{k-\frac{1}{k}}{\frac{k+1}{k}}$

$k-1$

$\frac{\frac{m}{m+1}-1}{\frac{m+1}{2}}$

$\frac{\frac{2xy}{2x-y}-y}{\frac{2x-y}{3}}$

$\frac{3{y}^{2}}{{\left(2x-y\right)}^{2}}$

$\frac{\frac{1}{a+b}-\frac{1}{a-b}}{\frac{1}{a+b}+\frac{1}{a-b}}$

$\frac{\frac{5}{x+3}-\frac{5}{x-3}}{\frac{5}{x+3}+\frac{5}{x-3}}$

$\frac{-3}{x}$

$\frac{2+\frac{1}{y+1}}{\frac{1}{y}+\frac{2}{3}}$

$\frac{\frac{1}{{x}^{2}}-\frac{1}{{y}^{2}}}{\frac{1}{x}+\frac{1}{y}}$

$\frac{y-x}{xy}$

$\frac{1+\frac{5}{x}+\frac{6}{{x}^{2}}}{1-\frac{1}{x}-\frac{12}{{x}^{2}}}$

$\frac{1+\frac{1}{y}-\frac{2}{{y}^{2}}}{1+\frac{7}{y}+\frac{10}{{y}^{2}}}$

$\frac{y-1}{y+5}$

$\frac{\frac{3n}{m}-2-\frac{m}{n}}{\frac{3n}{m}+4+\frac{m}{n}}$

$\frac{x-\frac{4}{3x-1}}{1-\frac{2x-2}{3x-1}}$

$3x-4$

$\frac{\frac{y}{x+y}-\frac{x}{x-y}}{\frac{x}{x+y}+\frac{y}{x-y}}$

$\frac{\frac{a}{a-2}-\frac{a}{a+2}}{\frac{2a}{a-2}+\frac{{a}^{2}}{a+2}}$

$\frac{4}{{a}^{2}+4}$

$3-\frac{2}{1-\frac{1}{m+1}}$

$\frac{x-\frac{1}{1-\frac{1}{x}}}{x+\frac{1}{1+\frac{1}{x}}}$

$\frac{\left(x-2\right)\left(x+1\right)}{\left(x-1\right)\left(x+2\right)}$

In electricity theory, when two resistors of resistance ${R}_{1}$ and ${R}_{2}$ ohms are connected in parallel, the total resistance $R$ is

$R=\frac{1}{\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}}$

Write this complex fraction as a simple fraction.

According to Einstein’s theory of relativity, two velocities ${v}_{1}$ and ${v}_{2}$ are not added according to $v={v}_{1}+{v}_{2}$ , but rather by

$v=\frac{{v}_{1}+{v}_{2}}{1+\frac{{v}_{1}{v}_{2}}{{c}^{2}}}$

Write this complex fraction as a simple fraction.

Einstein's formula is really only applicale for velocities near the speed of light $\left(c=186,000\text{\hspace{0.17em}}\text{miles\hspace{0.17em}per\hspace{0.17em}second}\right).$ At very much lower velocities, such as 500 miles per hour, the formula $v={v}_{1}+{v}_{2}$ provides an extremely good approximation.

$\frac{{c}^{2}\left({V}_{1}+{V}_{2}\right)}{{c}^{2}+{V}_{1}{V}_{2}}$

## Exercises for review

( [link] ) Supply the missing word. Absolute value speaks to the question of how and not “which way.”

( [link] ) Find the product. ${\left(3x+4\right)}^{2}.$

$9{x}^{2}+24x+16$

( [link] ) Factor ${x}^{4}-{y}^{4}.$

( [link] ) Solve the equation $\frac{3}{x-1}-\frac{5}{x+3}=0.$

$x=7$

( [link] ) One inlet pipe can fill a tank in 10 minutes. Another inlet pipe can fill the same tank in 4 minutes. How long does it take both pipes working together to fill the tank?

#### Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.