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Consider the fraction 6 24 . Multiply this fraction by 1. This is written 6 24 · 1 . But 1 can be rewritten as 1 6 1 6 .

6 24 1 6 1 6 = 6 1 6 24 1 6 = 1 4

The answer, 1 4 , is the reduced form. Notice that in 1 4 there is no factor common to both the numerator and denominator. This reasoning provides justification for the following rule.

Cancelling

Multiplying or dividing the numerator and denominator by the same nonzero number does not change the value of a fraction.

The process

We can now state a process for reducing a rational expression.

Reducing a rational expression

  1. Factor the numerator and denominator completely.
  2. Divide the numerator and denominator by all factors they have in common, that is, remove all factors of 1.

Reduced to lowest terms

  1. A rational expression is said to be reduced to lowest terms when the numerator and denominator have no factors in common.

Sample set a

Reduce the following rational expressions.

15 x 20 x . Factor . 15 x 20 x = 5 · 3 · x 5 · 2 · 2 · x The factors that are common to both the numerator and denominator are 5 and   x . Divide each by  5 x . 5 · 3 · x 5 · 2 · 2 · x = 3 4 , x 0 It is helpful to draw a line through the divided-out factors .

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x 2 4 x 2 6 x + 8 . Factor. ( x + 2 ) ( x 2 ) ( x 2 ) ( x 4 ) The factor that is common to both the numerator and denominator is  x 2.  Divide each by  x 2. ( x + 2 ) ( x 2 ) ( x 2 ) ( x 4 ) = x + 2 x 4 , x 2 , 4

The expression x 2 x 4 is the reduced form since there are no factors common to both the numerator and denominator. Although there is an x in both, it is a common term , not a common factor , and therefore cannot be divided out.

CAUTION — This is a common error: x 2 x 4 = x 2 x 4 = 2 4 is incorrect!

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a + 2 b 6 a + 12 b . Factor . a + 2 b 6 ( a + 2 b ) = a + 2 b 6 ( a + 2 b ) = 1 6 , a - 2 b
Since a + 2 b is a common factor to both the numerator and denominator, we divide both by a + 2 b . Since ( a + 2 b ) ( a + 2 b ) = 1 , we get 1 in the numerator.

Sometimes we may reduce a rational expression by using the division rule of exponents.

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8 x 2 y 5 4 x y 2 . Factor and use the rule  a n a m = a n m . 8 x 2 y 5 4 x y 2 = 2 2 2 2 2 x 2 1 y 5 2 = 2 x y 3 , x 0 , y 0

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10 x 3 a ( x 2 36 ) 2 x 3 10 x 2 12 x . Factor . 10 x 3 a ( x 2 36 ) 2 x 3 10 x 2 12 x = 5 2 x 3 a ( x + 6 ) ( x 6 ) 2 x ( x 2 5 x 6 ) = 5 2 x 3 a ( x + 6 ) ( x 6 ) 2 x ( x 6 ) ( x + 1 ) = 5 2 x 2 3 a ( x + 6 ) ( x 6 ) 2 x ( x 6 ) ( x + 1 ) = 5 x 2 a ( x + 6 ) x 1 , x 1 , 6

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x 2 x 12 x 2 + 2 x + 8 . Since it is most convenient to have the leading terms of a polynomial positive, factor out 1 from the denominator . x 2 x 12 ( x 2 2 x 8 ) Rewrite this . x 2 x 12 x 2 2 x 8 Factor . ( x 4 ) ( x + 3 ) ( x 4 ) ( x + 2 ) x + 3 x + 2 = ( x + 3 ) x + 2 = x 3 x + 2 , x 2 , 4

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a b b a . The numerator and denominator have the same terms but they occur with opposite signs . Factor 1 from the denominator . a b ( b + a ) = a b ( a b ) = a b a b = 1 , a b

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Practice set a

Reduce each of the following fractions to lowest terms.

x 2 9 x 2 + 5 x + 6

x 3 x + 2

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18 a 3 b 5 c 7 3 a b 3 c 5

6 a 2 b 2 c 2

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3 a 4 + 75 a 2 2 a 3 16 a 2 + 30 a

3 a ( a + 5 ) 2 ( a 3 )

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x 2 5 x + 4 x 2 + 12 x 32

x + 1 x 8

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2 x y y 2 x

−1

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Excercises

For the following problems, reduce each rational expression to lowest terms.

6 3 x - 12

2 ( x 4 )

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9 3 y - 21

3 ( y 7 )

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7 7 x - 14

1 ( x 2 )

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16 a 2 b 3 2 a b 2

8 a b

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( x + 3 ) ( x - 2 ) ( x + 3 ) ( x + 5 )

x 2 x + 5

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( y - 1 ) ( y - 7 ) ( y - 1 ) ( y + 6 )

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( a + 6 ) ( a - 5 ) ( a - 5 ) ( a + 2 )

a + 6 a + 2

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( m - 3 ) ( m - 1 ) ( m - 1 ) ( m + 4 )

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( y - 2 ) ( y - 3 ) ( y - 3 ) ( y - 2 )

1

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( x + 7 ) ( x + 8 ) ( x + 8 ) ( x + 7 )

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- 12 x 2 ( x + 4 ) 4 x

3 x ( x + 4 )

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- 3 a 4 ( a - 1 ) ( a + 5 ) - 2 a 3 ( a - 1 ) ( a + 9 )

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6 x 2 y 5 ( x - 1 ) ( x + 4 ) - 2 x y ( x + 4 )

3 x y 4 ( x 1 )

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22 a 4 b 6 c 7 ( a + 2 ) ( a - 7 ) 4 c ( a + 2 ) ( a - 5 )

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( x + 10 ) 3 x + 10

( x + 10 ) 2

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( x - 8 ) 2 ( x + 6 ) 4 ( x - 8 ) ( x + 6 )

( x 8 ) ( x + 6 ) 3

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( a + 1 ) 5 ( a - 1 ) 7 ( a + 1 ) 3 ( a - 1 ) 4

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( y - 2 ) 6 ( y - 1 ) 4 ( y - 2 ) 3 ( y - 1 ) 2

( y 2 ) 3 ( y 1 ) 2

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( x + 10 ) 5 ( x - 6 ) 3 ( x - 6 ) ( x + 10 ) 2

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( a + 6 ) 2 ( a - 7 ) 6 ( a + 6 ) 5 ( a - 7 ) 2

( a 7 ) 4 ( a + 6 ) 3

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( m + 7 ) 4 ( m - 8 ) 5 ( m + 7 ) 7 ( m - 8 ) 2

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( a + 2 ) ( a - 1 ) 3 ( a + 1 ) ( a - 1 )

( a + 2 ) ( a 1 ) 2 ( a + 1 )

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( b + 6 ) ( b - 2 ) 4 ( b - 1 ) ( b - 2 )

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8 ( x + 2 ) 3 ( x - 5 ) 6 2 ( x + 2 ) ( x - 5 ) 2

4 ( x + 2 ) 2 ( x 5 ) 4

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14 ( x - 4 ) 3 ( x - 10 ) 6 - 7 ( x - 4 ) 2 ( x - 10 ) 2

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x 2 + x - 12 x 2 - 4 x + 3

( x + 4 ) ( x 1 )

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x 2 + 3 x - 10 x 2 + 2 x - 15

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x 2 - 10 x + 21 x 2 - 6 x - 7

( x 3 ) ( x + 1 )

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x 2 + 10 x + 24 x 2 + 6 x

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x 2 + 9 x + 14 x 2 + 7 x

( x + 2 ) x

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6 b 2 - b 6 b 2 + 11 b - 2

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3 b 2 + 10 b + 3 3 b 2 + 7 b + 2

b + 3 b + 2

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4 b 2 - 1 2 b 2 + 5 b - 3

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16 a 2 - 9 4 a 2 - a - 3

( 4 a 3 ) ( a 1 )

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20 x 2 + 28 x y + 9 y 2 4 x 2 + 4 x y + y 2

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For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms.

x + 3 x + 4

( x + 3 ) ( x + 4 )

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5 a - 5 - 5

( a 1 ) or a + 1

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8 x - 16 - 4

2 ( x 2 )

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- 3 x + 10 10

3 x + 10 10

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a 6 - a 4 a 3

a ( a + 1 ) ( a 1 )

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x 4 x 5 - 3 x

x 3 x 4 3

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Excercises for review

( [link] ) Write ( 4 4 a 8 b 10 4 2 a 6 b 2 ) 1 so that only positive exponents appear.

1 16 a 2 b 8

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( [link] ) Factor 10 x 2 17 x + 3 .

( 5 x 1 ) ( 2 x 3 )

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( [link] ) Supply the missing word. An equation expressed in the form a x + b y = c is said to be expressed in form.

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( [link] ) Find the domain of the rational expression 2 x 2 3 x 18 .

x 3 , 6

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
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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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