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


  • 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
x 2 - 4 x 2 - 6 x + 8
can be reduced to the simpler expression x + 2 x - 4 for all x except x = 2 , 4 .

From our discussion of equality of fractions in Section [link] , we know that a b = c d when a d = b c . This fact allows us to deduce that, if k 0 , a k b k = a b , since a k b = a b k (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.
a k b k = a k b k = a b


The process of removing common factors is commonly called cancelling .

16 40 can be reduced to 2 5 .   Process:

16 40 = 2 · 2 · 2 · 2 2 · 2 · 2 · 5

Remove the three factors of 1; 2 2 · 2 2 · 2 2 .

2 · 2 · 2 · 2 2 · 2 · 2 · 5 = 2 5

Notice that in 2 5 , there is no factor common to the numerator and denominator.

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111 148 can be reduced to 3 4 .   Process:

111 148 = 3 · 37 4 · 37

Remove the factor of 1; 37 37 .

3 · 37 4 · 37

3 4

Notice that in 3 4 , there is no factor common to the numerator and denominator.

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3 9 can be reduced to 1 3 .   Process:

3 9 = 3 · 1 3 · 3

Remove the factor of 1; 3 3 .

3 · 1 3 · 3 = 1 3

Notice that in 1 3 there is no factor common to the numerator and denominator.

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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.

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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.

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian Reply
absolutely yes
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Akash Reply
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characteristics of micro business
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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