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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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.
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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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