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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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele 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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