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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

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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
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.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
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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