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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

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
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
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.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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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