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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. Therefore, no work space is provided, nor does the chapter contain all of the pedagogical features of the text. As a review, this chapter can be assigned at the discretion of the instructor and can also be a valuable reference tool for the student.

Overview

  • Factors
  • Exponential Notation

Factors

Let’s begin our review of arithmetic by recalling the meaning of multiplication for whole numbers (the counting numbers and zero).

Multiplication

Multiplication is a description of repeated addition.

In the addition

7 + 7 + 7 + 7

the number 7 is repeated as an addend* 4 times. Therefore, we say we have four times seven and describe it by writing

4 · 7

The raised dot between the numbers 4 and 7 indicates multiplication. The dot directs us to multiply the two numbers that it separates. In algebra, the dot is preferred over the symbol × to denote multiplication because the letter x is often used to represent a number. Thus,

4 · 7 = 7 + 7 + 7 + 7

Factors and products

In a multiplication, the numbers being multiplied are called factors. The result of a multiplication is called the product. For example, in the multiplication

4 · 7 = 28

the numbers 4 and 7 are factors, and the number 28 is the product. We say that 4 and 7 are factors of 28. (They are not the only factors of 28. Can you think of others?)

Now we know that

( factor ) · ( factor ) = product

This indicates that a first number is a factor of a second number if the first number divides into the second number with no remainder. For example, since

4 · 7 = 28

both 4 and 7 are factors of 28 since both 4 and 7 divide into 28 with no remainder.

Exponential notation

Quite often, a particular number will be repeated as a factor in a multiplication. For example, in the multiplication

7 · 7 · 7 · 7

the number 7 is repeated as a factor 4 times. We describe this by writing 7 4 . Thus,

7 · 7 · 7 · 7 = 7 4

The repeated factor is the lower number (the base), and the number recording how many times the factor is repeated is the higher number (the superscript). The superscript number is called an exponent.

Exponent

An exponent is a number that records how many times the number to which it is attached occurs as a factor in a multiplication.

Sample set a

For Examples 1, 2, and 3, express each product using exponents.

3 · 3 · 3 · 3 · 3 · 3.   Since 3 occurs as a factor 6 times,

3 · 3 · 3 · 3 · 3 · 3 = 3 6

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8 · 8.   Since 8 occurs as a factor 2 times,

8 · 8 = 8 2

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5 · 5 · 5 · 9 · 9.   Since 5 occurs as a factor 3 times, we have 5 3 . Since 9 occurs as a factor 2 times, we have 9 2 . We should see the following replacements.

5 · 5 · 5 5 3 · 9 · 9 9 2
Then we have

5 · 5 · 5 · 9 · 9 = 5 3 · 9 2

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Expand 3 5 .   The base is 3 so it is the repeated factor. The exponent is 5 and it records the number of times the base 3 is repeated. Thus, 3 is to be repeated as a factor 5 times.

3 5 = 3 · 3 · 3 · 3 · 3

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Expand 6 2 · 10 4 .   The notation 6 2 · 10 4 records the following two facts: 6 is to be repeated as a factor 2 times and 10 is to be repeated as a factor 4 times. Thus,

6 2 · 10 4 = 6 · 6 · 10 · 10 · 10 · 10

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Exercises

For the following problems, express each product using exponents.

8 · 8 · 8

8 3

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12 · 12 · 12 · 12 · 12

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5 · 5 · 5 · 5 · 5 · 5 · 5

5 7

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3 · 3 · 3 · 3 · 3 · 4 · 4

3 5 · 4 2

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8 · 8 · 8 · 15 · 15 · 15 · 15

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2 · 2 · 2 · 9 · 9 · 9 · 9 · 9 · 9 · 9 · 9

2 3 · 9 8

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3 · 3 · 10 · 10 · 10

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Suppose that the letters x and y are each used to represent numbers. Use exponents to express the following product.

x · x · x · y · y

x 3 · y 2

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Suppose that the letters x and y are each used to represent numbers. Use exponents to express the following product.

x · x · x · x · x · y · y · y

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For the following problems, expand each product (do not compute the actual value).

3 4

3 · 3 · 3 · 3

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2 5

2 · 2 · 2 · 2 · 2

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5 3 · 6 2

5 · 5 · 5 · 6 · 6

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x 4 · y 4

x · x · x · x · y · y · y · y

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For the following problems, specify all the whole number factors of each number. For example, the complete set of whole number factors of 6 is 1, 2, 3, 6.

20

1 , 2 , 4 , 5 , 10 , 20

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12

1 , 2 , 3 , 4 , 6 , 12

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21

1 , 3 , 7 , 21

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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 .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
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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