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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy.Objectives of this module: be able to use the division property of square roots, the method of rationalizing the denominator, and conjugates to divide square roots.

Overview

  • The Division Property of Square Roots
  • Rationalizing the Denominator
  • Conjugates and Rationalizing the Denominator

The division property of square roots

In our work with simplifying square root expressions, we noted that

x y = x y

Since this is an equation, we may write it as

x y = x y

To divide two square root expressions, we use the division property of square roots.

The division property x y = x y

x y = x y

The quotient of the square roots is the square root of the quotient.

Rationalizing the denominator

As we can see by observing the right side of the equation governing the division of square roots, the process may produce a fraction in the radicand. This means, of course, that the square root expression is not in simplified form. It is sometimes more useful to rationalize the denominator of a square root expression before actually performing the division.

Sample set a

Simplify the square root expressions.

3 7 .

This radical expression is not in simplified form since there is a fraction under the radical sign. We can eliminate this problem using the division property of square roots.

3 7 = 3 7 = 3 7 · 7 7 = 3 7 7 = 21 7

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5 3 .

A direct application of the rule produces 5 3 , which must be simplified. Let us rationalize the denominator before we perform the division.

5 3 = 5 3 · 3 3 = 5 3 3 = 15 3

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21 7 = 21 7 = 3 .

The rule produces the quotient quickly. We could also rationalize the denominator first and produce the same result.

21 7 = 21 7 · 7 7 = 21 · 7 7 = 3 · 7 · 7 7 = 3 · 7 2 7 = 7 3 7 = 3

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80 x 9 5 x 4 = 80 x 9 5 x 4 = 16 x 5 = 16 x 4 x = 4 x 2 x

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50 a 3 b 7 5 a b 5 = 50 a 3 b 7 5 a b 5 = 10 a 2 b 2 = a b 10

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5 a b .

Some observation shows that a direct division of the radicands will produce a fraction. This suggests that we rationalize the denominator first.

5 a b = 5 a b · b b = 5 a b b = 5 a b b

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m - 6 m + 2 = m - 6 m + 2 · m + 2 m + 2 = m 2 - 4 m - 12 m + 2

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y 2 - y - 12 y + 3 = y 2 - y - 12 y + 3 = ( y + 3 ) ( y - 4 ) ( y + 3 ) = ( y + 3 ) ( y - 4 ) ( y + 3 ) = y - 4

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Practice set a

Simplify the square root expressions.

80 m 5 n 8 5 m 2 n

4 m n 3 m n

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196 ( x + 7 ) 8 2 ( x + 7 ) 3

7 ( x + 7 ) 2 2 ( x + 7 )

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n + 4 n - 5

n 2 - n - 20 n - 5

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a 2 - 6 a + 8 a - 2

a - 4

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a 3 m - 5 a m - 1

a m - 2

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Conjugates and rationalizing the denominator

To perform a division that contains a binomial in the denominator, such as 3 4 + 6 , we multiply the numerator and denominator by a conjugate of the denominator.

Conjugate

A conjugate of the binomial a + b is a - b . Similarly, a conjugate of a - b is a + b .

Notice that when the conjugates a + b and a - b are multiplied together, they produce a difference of two squares.

( a + b ) ( a - b ) = a 2 - a b + a b - b 2 = a 2 - b 2

This principle helps us eliminate square root radicals, as shown in these examples that illustrate finding the product of conjugates.

( 5 + 2 ) ( 5 - 2 ) = 5 2 - ( 2 ) 2 = 25 - 2 = 23

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( 6 - 7 ) ( 6 + 7 ) = ( 6 ) 2 - ( 7 ) 2 = 6 - 7 = - 1

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Sample set b

Simplify the following expressions.

3 4 + 6 .

The conjugate of the denominator is 4 - 6. Multiply the fraction by 1 in the form of 4 - 6 4 - 6 . 3 4 + 6 · 4 - 6 4 - 6 = 3 ( 4 - 6 ) 4 2 - ( 6 ) 2 = 12 - 3 6 16 - 6 = 12 - 3 6 10

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2 x 3 - 5 x .

The conjugate of the denominator is 3 + 5 x . Multiply the fraction by 1 in the form of 3 + 5 x 3 + 5 x .

2 x 3 5 x · 3 + 5 x 3 + 5 x = 2 x ( 3 + 5 x ) ( 3 ) 2 ( 5 x ) 2 = 2 x 3 + 2 x 5 x 3 5 x = 6 x + 10 x 2 3 5 x = 6 x + x 10 3 5 x

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Practice set b

Simplify the following expressions.

- 2 1 - 3 x

- 2 - 2 3 x 1 - 3 x

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8 3 x + 2 x

2 6 x - 4 x x

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2 m m - 3 m

2 m + 6 m - 3

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Exercises

For the following problems, simplify each expressions.

45 a 3 b 8 c 2 5 a b 2 c

3 a b 3 c

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30 p 5 q 14 5 q 7

p 2 q 3 6 p q

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3 m 4 n 3 6 m n 5

m 2 m 2 n

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5 ( p - q ) 6 ( r + s ) 4 25 ( r + s ) 3

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m ( m - 6 ) - m 2 + 6 m 3 m - 7

0

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s + 3 s - 3

s 2 9 s 3

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x 2 - 10 x + 24 x - 4

x 6

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x 2 - 4 x + 3 x - 3

x 1

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- 5 4 + 5

20 + 5 5 11

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2 1 - a

2 ( 1 + a ) 1 a

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- 6 7 + 2

2 ( 7 2 )

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6 y 1 + 3 y

6 y 3 y 2 1 3 y

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a a + b

a a b a b

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Exercises for review

( [link] ) Simplify x 8 y 7 ( x 4 y 8 x 3 y 4 ) .

x 9 y 11

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( [link] ) Solve the compound inequality 8 7 5 x 23.

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( [link] ) Construct the graph of y = 2 3 x - 4.
An xy-plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through two points with coordinates three, negative two; and zero, negative five.

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( [link] ) The symbol x represents which square root of the number x , x 0 ?

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( [link] ) Simplify a 2 + 8 a + 16 .

a + 4

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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, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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