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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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what are the types of wave
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answer
Magreth
progressive wave
Magreth
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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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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