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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

where we get a research paper on Nano chemistry....?
Maira Reply
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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