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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret.Objectives of this module: understand exponential notation, be able to read exponential notation, understand how to use exponential notation with the order of operations.


  • Exponential Notation
  • Reading Exponential Notation
  • The Order of Operations

Exponential notation

In Section [link] we were reminded that multiplication is a description for repeated addition. A natural question is “Is there a description for repeated multiplication?” The answer is yes. The notation that describes repeated multiplication is exponential notation .


In multiplication, the numbers being multiplied together are called factors . In repeated multiplication, all the factors are the same. In nonrepeated multiplication, none of the factors are the same. For example,

18 18 18 18 Repeated multiplication of 18. All four factors , 18 , are the same . x x x x x Repeated multiplication of x . All five factors , x , are the same . 3 7 a Nonrepeated multiplication . None of the factors are the same .

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Exponential notation is used to show repeated multiplication of the same factor. The notation consists of using a superscript on the factor that is repeated . The superscript is called an exponent .

Exponential notation

If x is any real number and n is a natural number, then

x n = x x x ... x n factors of x

An exponent records the number of identical factors in a multiplication.

Note that the definition for exponential notation only has meaning for natural number exponents. We will extend this notation to include other numbers as exponents later.

Sample set a

7 7 7 7 7 7 = 7 6 .

The repeated factor is 7. The exponent 6 records the fact that 7 appears 6 times in the multiplication.

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x x x x = x 4 .

The repeated factor is x . The exponent 4 records the fact that x appears 4 times in the multiplication.

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( 2 y ) ( 2 y ) ( 2 y ) = ( 2 y ) 3 .

The repeated factor is 2 y . The exponent 3 records the fact that the factor 2 y appears 3 times in the multiplication.

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2 y y y = 2 y 3 .

The repeated factor is y . The exponent 3 records the fact that the factor y appears 3 times in the multiplication.

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( a + b ) ( a + b ) ( a b ) ( a b ) ( a b ) = ( a + b ) 2 ( a b ) 3 .

The repeated factors are ( a + b ) and ( a b ) , ( a + b ) appearing 2 times and ( a b ) appearing 3 times.

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Practice set a

Write each of the following using exponents.

( 3 b ) ( 3 b ) ( 5 c ) ( 5 c ) ( 5 c ) ( 5 c )

( 3 b ) 2 ( 5 c ) 4

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2 2 7 7 7 ( a 4 ) ( a 4 )

2 2 7 3 ( a 4 ) 2

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8 x x x y z z z z z

8 x 3 y z 5

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It is extremely important to realize and remember that an exponent applies only to the factor to which it is directly connected.

Sample set b

8 x 3 means 8 x x x and not 8 x 8 x 8 x . The exponent 3 applies only to the factor x since it is only to the factor x that the 3 is connected.

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Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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.
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
How can I make nanorobot?
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
how can I make nanorobot?
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
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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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