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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 identify a perfect square, be familiar with the product and quotient properties of square roots, be able to simplify square roots involving and not involving fractions.


  • Perfect Squares
  • The Product Property of Square Roots
  • The Quotient Property of Square Roots
  • Square Roots Not Involving Fractions
  • Square Roots Involving Fractions

To begin our study of the process of simplifying a square root expression, we must note three facts: one fact concerning perfect squares and two concerning properties of square roots.

Perfect squares

Perfect squares

Real numbers that are squares of rational numbers are called perfect squares. The numbers 25 and 1 4 are examples of perfect squares since 25 = 5 2 and 1 4 = ( 1 2 ) 2 , and 5 and 1 2 are rational numbers. The number 2 is not a perfect square since 2 = ( 2 ) 2 and 2 is not a rational number.

Although we will not make a detailed study of irrational numbers, we will make the following observation:

Any indicated square root whose radicand is not a perfect square is an irrational number.

The numbers 6 , 15 , and 3 4 are each irrational since each radicand ( 6 , 15 , 3 4 ) is not a perfect square.

The product property of square roots

Notice that

9 · 4 = 36 = 6      and
9 4 = 3 · 2 = 6

Since both 9 · 4 and 9 4 equal 6, it must be that

9 · 4 = 9 4

The product property x y = x y

This suggests that in general, if x and y are positive real numbers,

x y = x y

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

The quotient property of square roots

We can suggest a similar rule for quotients. Notice that

36 4 = 9 = 3      and
36 4 = 6 2 = 3

Since both 36 4 and 36 4 equal 3, it must be that

36 4 = 36 4

The quotient property x y = x y

This suggests that in general, if x and y are positive real numbers,

x y = x y ,       y 0

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

It is extremely important to remember that

x + y x + y or x y x y

For example, notice that 16 + 9 = 25 = 5 , but 16 + 9 = 4 + 3 = 7.

We shall study the process of simplifying a square root expression by distinguishing between two types of square roots: square roots not involving a fraction and square roots involving a fraction.

Square roots not involving fractions

A square root that does not involve fractions is in simplified form if there are no perfect square in the radicand.

The square roots x , a b , 5 m n , 2 ( a + 5 ) are in simplified form since none of the radicands contains a perfect square.

The square roots x 2 , a 3 = a 2 a are not in simplified form since each radicand contains a perfect square.

Questions & Answers

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absolutely yes
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
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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.
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s. Reply
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Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Sanket Reply
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
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what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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silver nanoparticles could handle the job?
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Source:  OpenStax, Algebra ii for the community college. OpenStax CNX. Jul 03, 2014 Download for free at http://cnx.org/content/col11671/1.1
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