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This module provides techniques for simplifying rational expressions.

How do you simplify a fraction? The answer is, you divide the top and bottom by the same thing.

4 6 = 4 ÷ 2 6 ÷ 2 = 2 3 size 12{ { {4} over {6} } = { {4 div 2} over {6 div 2} } = { {2} over {3} } } {}

So 4 6 size 12{ { {4} over {6} } } {} and 2 3 size 12{ { {2} over {3} } } {} are two different ways of writing the same number.

4/6 of a Pie cut into 6 equal pieces On the left, a pizza divided into six equal slices: the four shaded-in regions represent 4 6 size 12{ { {4} over {6} } } {} of a pizza. On the right, a pizza divided into three equal slices: the two shaded-in regions represent 2 3 size 12{ { {2} over {3} } } {} of a pizza. The two areas are identical: 4 6 size 12{ { {4} over {6} } } {} and 2 3 size 12{ { {2} over {3} } } {} are two different ways of expressing the same amount of pizza . 2/3 of Pie cut into 3 pieces.

In some cases, you have to repeat this process more than once before the fraction is fully simplified.

40 48 = 40 ÷ 4 48 ÷ 4 = 10 12 = 10 ÷ 2 12 ÷ 2 = 5 6 size 12{ { {"40"} over {"48"} } = { {"40" div 4} over {"48" div 4} } = { {"10"} over {"12"} } = { {"10" div 2} over {"12" div 2} } = { {5} over {6} } } {}

It is vital to remember that we have not divided this fraction by 4, or by 2, or by 8 . We have rewritten the fraction in another form: 40 48 size 12{ { {"40"} over {"48"} } } {} is the same number as 5 6 size 12{ { {5} over {6} } } {} . In strictly practical terms, if you are given the choice between 40 48 size 12{ { {"40"} over {"48"} } } {} of a pizza or 5 6 size 12{ { {5} over {6} } } {} of a pizza, it does not matter which one you choose, because they are the same amount of pizza.

You can divide the top and bottom of a fraction by the same number, but you cannot subtract the same number from the top and bottom of a fraction!

40 48 = 40 39 48 39 = 1 9 size 12{ { {"40"} over {"48"} } = { {"40" - "39"} over {"48" - "39"} } = { {1} over {9} } } {} Wrong!

Given the choice, a hungry person would be wise to choose 40 48 size 12{ { {"40"} over {"48"} } } {} of a pizza instead of 1 9 size 12{ { {1} over {9} } } {} .

Dividing the top and bottom of a fraction by the same number leaves the fraction unchanged, and that is how you simplify fractions. Subtracting the same number from the top and bottom changes the value of the fraction, and is therefore an illegal simplification.

All this is review. But if you understand these basic fraction concepts, you are ahead of many Algebra II students! And if you can apply these same concepts when variables are involved , then you are ready to simplify rational expressions, because there are no new concepts involved.

As an example, consider the following:

x 2 9 x 2 + 6x + 9 size 12{ { {x rSup { size 8{2} } - 9} over {x rSup { size 8{2} } +6x+9} } } {}

You might at first be tempted to cancel the common x 2 size 12{x rSup { size 8{2} } } {} terms on the top and bottom. But this would be, mathematically, subtracting x 2 size 12{x rSup { size 8{2} } } {} from both the top and the bottom; which, as we have seen, is an illegal fraction operation.

x 2 9 x 2 + 6x + 9 size 12{ { {x rSup { size 8{2} } - 9} over {x rSup { size 8{2} } +6x+9} } } {} = 9 6x + 9 size 12{ {}= { { - 9} over {6x+9} } } {} Wrong!

To properly simplify this expression, begin by factoring both the top and the bottom, and then see if anything cancels.

Simplifying rational expressions

x 2 9 x 2 + 6x + 9 size 12{ { {x rSup { size 8{2} } - 9} over {x rSup { size 8{2} } +6x+9} } } {} The problem
= ( x + 3 ) ( x 3 ) ( x + 3 ) 2 size 12{ {}= { { \( x+3 \) \( x - 3 \) } over { \( x+3 \) rSup { size 8{2} } } } } {} Always begin rational expression problems by factoring! This factors easily, thanks to ( x + a ) ( x a ) = x 2 a x size 12{ \( x+a \) \( x - a \) =x rSup { size 8{2} } - a rSup { size 8{x} } } {} and ( x + a ) 2 = x 2 + 2 ax + a 2 size 12{ \( x+a \) rSup { size 8{2} } =x rSup { size 8{2} } +2 ital "ax"+a rSup { size 8{2} } } {}
= x 3 x + 3 size 12{ {}= { {x - 3} over {x+3} } } {} Cancel a common ( x + 3 ) size 12{ \( x+3 \) } {} term on both the top and the bottom. This is legal because this term was multiplied on both top and bottom; so we are effectively dividing the top and bottom by ( x + 3 ) size 12{ \( x+3 \) } {} , which leaves the fraction unchanged .

What we have created, of course, is an algebraic generalization:

x 2 9 x 2 + 6x + 9 = x 3 x + 3 size 12{ { {x rSup { size 8{2} } - 9} over {x rSup { size 8{2} } +6x+9} } = { {x - 3} over {x+3} } } {}

For any x value, the complicated expression on the left will give the same answer as the much simpler expression on the right. You may want to try one or two values, just to confirm that it works.

As you can see, the skills of factoring and simplifying fractions come together in this exercise. No new skills are required.

Questions & Answers

what are the products of Nano chemistry?
Maira Reply
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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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Source:  OpenStax, Rational expressions. OpenStax CNX. Feb 28, 2011 Download for free at http://cnx.org/content/col11278/1.2
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