# 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.

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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.

#### Get Jobilize Job Search Mobile App in your pocket Now! By By By Stephen Voron By Yasser Ibrahim By By Madison Christian By OpenStax By Ann Schlosser By OpenStax By Richley Crapo By Rhodes By OpenStax