# 1.3 Arithmetic review: factors, products, and exponents

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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.

$7+7+7+7$

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

$4\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7$

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,

$4\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7=7+7+7+7$

## 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

$4\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7=28$

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

$\begin{array}{ccc}\left(\text{factor}\right)\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(\text{factor}\right)& =& \text{product}\end{array}$

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

$4\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7=28$

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

$7\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7$

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

$7\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7={7}^{4}$

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.

$3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3.$   Since 3 occurs as a factor 6 times,

$3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3={3}^{6}$

$8\text{\hspace{0.17em}}·\text{\hspace{0.17em}}8.$   Since 8 occurs as a factor 2 times,

$8\text{\hspace{0.17em}}·\text{\hspace{0.17em}}8={8}^{2}$

$5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9.$   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.

$\underset{{5}^{3}}{\underbrace{5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\underset{{9}^{2}}{\underbrace{9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9}}$
Then we have

$5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9={5}^{3}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{9}^{2}$

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.

${3}^{5}=3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3$

Expand ${6}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{10}^{4}.$   The notation ${6}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{10}^{4}$ 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,

${6}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{10}^{4}=6\text{\hspace{0.17em}}·\text{\hspace{0.17em}}6\text{\hspace{0.17em}}·\text{\hspace{0.17em}}10\text{\hspace{0.17em}}·\text{\hspace{0.17em}}10\text{\hspace{0.17em}}·\text{\hspace{0.17em}}10\text{\hspace{0.17em}}·\text{\hspace{0.17em}}10$

## Exercises

For the following problems, express each product using exponents.

$8\text{\hspace{0.17em}}·\text{\hspace{0.17em}}8\text{\hspace{0.17em}}·\text{\hspace{0.17em}}8$

${8}^{3}$

$12\text{\hspace{0.17em}}·\text{\hspace{0.17em}}12\text{\hspace{0.17em}}·\text{\hspace{0.17em}}12\text{\hspace{0.17em}}·\text{\hspace{0.17em}}12\text{\hspace{0.17em}}·\text{\hspace{0.17em}}12$

$5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5$

${5}^{7}$

$1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}1$

$3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}4\text{\hspace{0.17em}}·\text{\hspace{0.17em}}4$

${3}^{5}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{4}^{2}$

$8\text{\hspace{0.17em}}·\text{\hspace{0.17em}}8\text{\hspace{0.17em}}·\text{\hspace{0.17em}}8\text{\hspace{0.17em}}·\text{\hspace{0.17em}}15\text{\hspace{0.17em}}·\text{\hspace{0.17em}}15\text{\hspace{0.17em}}·\text{\hspace{0.17em}}15\text{\hspace{0.17em}}·\text{\hspace{0.17em}}15$

$2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9$

${2}^{3}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{9}^{8}$

$3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}10\text{\hspace{0.17em}}·\text{\hspace{0.17em}}10\text{\hspace{0.17em}}·\text{\hspace{0.17em}}10$

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

$x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}y\text{\hspace{0.17em}}·\text{\hspace{0.17em}}y$

${x}^{3}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{y}^{2}$

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

$x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}y\text{\hspace{0.17em}}·\text{\hspace{0.17em}}y\text{\hspace{0.17em}}·\text{\hspace{0.17em}}y$

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

${3}^{4}$

$3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3$

${4}^{3}$

${2}^{5}$

$2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2$

${9}^{6}$

${5}^{3}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{6}^{2}$

$5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}6\text{\hspace{0.17em}}·\text{\hspace{0.17em}}6$

${2}^{7}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{3}^{4}$

${x}^{4}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{y}^{4}$

$x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x\text{\hspace{0.17em}}·\text{\hspace{0.17em}}y\text{\hspace{0.17em}}·\text{\hspace{0.17em}}y\text{\hspace{0.17em}}·\text{\hspace{0.17em}}y\text{\hspace{0.17em}}·\text{\hspace{0.17em}}y$

${x}^{6}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{y}^{2}$

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

$1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}4,\text{\hspace{0.17em}}5,\text{\hspace{0.17em}}10,\text{\hspace{0.17em}}20$

14

12

$1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4,\text{\hspace{0.17em}}6,\text{\hspace{0.17em}}12$

30

21

$1,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}7,\text{\hspace{0.17em}}21$

45

11

$1,\text{\hspace{0.17em}}11$

17

19

$1,\text{\hspace{0.17em}}19$

2

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absolutely yes
Daniel
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Maciej
Abigail
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what is fullerene does it is used to make bukky balls
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
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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
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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Harper
Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Yasmin
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Cesar
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Uday
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
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Azam
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Prasenjit
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Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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