# 8.2 Reducing rational expressions

 Page 1 / 2
<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: understand and be able to use the process of reducing rational expressions.</para>

## Overview

• The Logic Behind The Process
• The Process

## The logic behind the process

When working with rational expressions, it is often best to write them in the simplest possible form. For example, the rational expression
$\frac{{x}^{2}-4}{{x}^{2}-6x+8}$
can be reduced to the simpler expression $\frac{x+2}{x-4}$ for all $x$ except $x=2,4$ .

From our discussion of equality of fractions in Section [link] , we know that $\frac{a}{b}=\frac{c}{d}$ when $ad=bc$ . This fact allows us to deduce that, if $k\ne 0,\frac{ak}{bk}=\frac{a}{b},$ since $akb=abk$ (recall the commutative property of multiplication). But this fact means that if a factor (in this case, $k$ ) is common to both the numerator and denominator of a fraction, we may remove it without changing the value of the fraction.
$\frac{ak}{bk}=\frac{a\overline{)k}}{b\overline{)k}}=\frac{a}{b}$

## Cancelling

The process of removing common factors is commonly called cancelling .

$\frac{16}{40}$ can be reduced to $\frac{2}{5}$ .   Process:

$\frac{16}{40}=\frac{2·2·2·2}{2·2·2·5}$

Remove the three factors of 1; $\frac{2}{2}·\frac{2}{2}·\frac{2}{2}.$

$\frac{\overline{)2}·\overline{)2}·\overline{)2}·2}{\overline{)2}·\overline{)2}·\overline{)2}·5}=\frac{2}{5}$

Notice that in $\frac{2}{5}$ , there is no factor common to the numerator and denominator.

$\frac{111}{148}$ can be reduced to $\frac{3}{4}$ .   Process:

$\frac{111}{148}=\frac{3·37}{4·37}$

Remove the factor of 1; $\frac{37}{37}$ .

$\frac{3·\overline{)37}}{4·\overline{)37}}$

$\frac{3}{4}$

Notice that in $\frac{3}{4}$ , there is no factor common to the numerator and denominator.

$\frac{3}{9}$ can be reduced to $\frac{1}{3}$ .   Process:

$\frac{3}{9}=\frac{3·1}{3·3}$

Remove the factor of 1; $\frac{3}{3}$ .

$\frac{\overline{)3}·1}{\overline{)3}·3}=\frac{1}{3}$

Notice that in $\frac{1}{3}$ there is no factor common to the numerator and denominator.

$\frac{5}{7}$ cannot be reduced since there are no factors common to the numerator and denominator.

Problems 1, 2, and 3 shown above could all be reduced. The process in each reduction included the following steps:

1. Both the numerator and denominator were factored.
2. Factors that were common to both the numerator and denominator were noted and removed by dividing them out.

We know that we can divide both sides of an equation by the same nonzero number, but why should we be able to divide both the numerator and denominator of a fraction by the same nonzero number? The reason is that any nonzero number divided by itself is 1, and that if a number is multiplied by 1, it is left unchanged.

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
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
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.