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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: understand the concept of square root, be able to distinguish between the principal and secondary square roots of a number, be able to relate square roots and meaningful expressions and to simplify a square root expression.


  • Square Roots
  • Principal and Secondary Square Roots
  • Meaningful Expressions
  • Simplifying Square Roots

Square roots

When we studied exponents in Section [link] , we noted that 4 2 = 16 and ( 4 ) 2 = 16. We can see that 16 is the square of both 4 and 4 . Since 16 comes from squaring 4 or 4 , 4 and 4 are called the square roots of 16. Thus 16 has two square roots, 4 and 4 . Notice that these two square roots are opposites of each other.

We can say that

Square root

The square root of a positive number x is a number such that when it is squared the number x results.

Every positive number has two square roots, one positive square root and one negative square root. Furthermore, the two square roots of a positive number are opposites of each other. The square root of 0 is 0.

Sample set a

The two square roots of 49 are 7 and −7 since

7 2 = 49 and ( 7 ) 2 = 49

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The two square roots of 49 64 are 7 8 and 7 8 since

( 7 8 ) 2 = 7 8 · 7 8 = 49 64 and ( 7 8 ) 2 = 7 8 · 7 8 = 49 64

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Practice set a

Name both square roots of each of the following numbers.

1 4

1 2 and  1 2

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9 16

3 4 and  3 4

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Principal and secondary square roots

There is a notation for distinguishing the positive square root of a number x from the negative square root of x .

Principal square root: x

If x is a positive real number, then

x represents the positive square root of x . The positive square root of a number is called the principal square root of the number.

Secondary square root: x

x represents the negative square root of x . The negative square root of a number is called the secondary square root of the number.

x indicates the secondary square root of x .

Radical sign, radicand, and radical

In the expression x ,

is called a radical sign .

x is called the radicand .

x is called a radical .

The horizontal bar that appears attached to the radical sign, , is a grouping symbol that specifies the radicand.

Because x and x are the two square root of x ,

( x ) ( x ) = x and ( x ) ( x ) = x

Sample set b

Write the principal and secondary square roots of each number.

9. Principal square root is  9 = 3. Secondary square root is 9 = 3.

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15. Principal square root is  15 . Secondary square root is 15 .

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Use a calculator to obtain a decimal approximation for the two square roots of 34. Round to two decimal places.
On the Calculater Type 34 Press x Display reads: 5.8309519 Round to 5.83.
Notice that the square root symbol on the calculator is . This means, of course, that a calculator will produce only the positive square root. We must supply the negative square root ourselves.

34 5.83 and 34 5.83
Note: The symbol ≈ means "approximately equal to."

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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
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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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