Download Exponents and Roots

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Exponents, Roots, and
Factorization of Whole Numbers:
Exponents and Roots
∗
Wade Ellis
Denny Burzynski
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0
†
Abstract
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr.
This
module discusses exponents and roots. By the end of the module students should be able to understand
and be able to read exponential notation, understand the concept of root and be able to read root
notation, and use a calculator having the
yx
key to determine a root.
1 Section Overview
•
•
•
•
•
Exponential Notation
Reading Exponential Notation
Roots
Reading Root Notation
Calculators
2 Exponential Notation
Exponential Notation
We have noted that multiplication is a description of repeated addition. Exponential notation is a
description of repeated multiplication.
Suppose we have the repeated multiplication
8·8·8·8·8
Exponent
The factor 8 is repeated 5 times. Exponential notation uses a superscript for the number of times the factor
is repeated. The superscript is placed on the repeated factor, 85 , in this case. The superscript is called an
exponent.
The Function of an Exponent
An exponent records the number
∗
†
of identical factors that are repeated in a multiplication.
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2.1 Sample Set A
Write the following multiplication using exponents.
Example 1
3 · 3. Since the factor 3 appears 2 times, we record this as
32
Example 2
62 · 62 · 62 · 62 · 62 · 62 · 62 · 62 · 62. Since the factor 62 appears 9 times, we record this as
629
Expand (write without exponents) each number.
Example 3
124 . The exponent 4 is recording 4 factors of 12 in a multiplication. Thus,
124 = 12 · 12 · 12 · 12
Example 4
7063 . The exponent 3 is recording 3 factors of 706 in a multiplication. Thus,
7063 = 706 · 706 · 706
2.2 Practice Set A
Write the following using exponents.
Exercise 1
(Solution on p. 10.)
Exercise 2
(Solution on p. 10.)
Exercise 3
(Solution on p. 10.)
37 · 37
16 · 16 · 16 · 16 · 16
9·9·9·9·9·9·9·9·9·9
Write each number without exponents.
Exercise 4
(Solution on p. 10.)
Exercise 5
(Solution on p. 10.)
853
47
Exercise 6
1, 7392
(Solution on p. 10.)
3 Reading Exponential Notation
In a number such as 85 ,
Base
8 is called the base.
Exponent, Power
5 is called the exponent,
or power. 85 is read as "eight to the fth power," or more simply as "eight to
the fth," or "the fth power of eight."
Squared
When a whole number is raised to the second power, it is said to be squared. The number 52 can be read
as
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5 to the second power, or
5 to the second, or
5 squared.
Cubed
When a whole number is raised to the third power, it is said to be cubed. The number 53 can be read as
5 to the third power, or
5 to the third, or
5 cubed.
When a whole number is raised to the power of 4 or higher, we simply say that that number is raised to
that particular power. The number 58 can be read as
5 to the eighth power, or just
5 to the eighth.
4 Roots
In the English language, the word "root" can mean a source of something. In mathematical terms, the word
"root" is used to indicate that one number is the source of another number through repeated multiplication.
Square Root
We know that 49 = 72 , that is, 49 = 7 · 7. Through repeated multiplication, 7 is the source of 49. Thus, 7 is
a root of 49. Since two 7's must be multiplied together to produce 49, the 7 is called the second or square
root of 49.
Cube Root
We know that 8 = 23 , that is, 8 = 2 · 2 · 2. Through repeated multiplication, 2 is the source of 8. Thus, 2 is a
root of 8. Since three 2's must be multiplied together to produce 8, 2 is called the third or cube root of 8.
We can continue this way to see such roots as fourth roots, fth roots, sixth roots, and so on.
5 Reading Root Notation
n
There is a symbol used to indicate roots of a number. It is called the radical sign √
The Radical Sign
√
n
√
n
The symbol
is called a radical sign and indicates the nth root of a number.
We discuss particular roots using the radical sign as follows:
Square
Root
√
number indicates the square root of the number under the radical sign. It is customary to drop the 2 in
the radical sign when discussing square roots. The symbol √
is understood to be the square root radical
sign.√
49 = 7 since 7 · 7 = 72 = 49
2
Cube
Root
√
3
number
indicates the cube root of the number under the radical sign.
√
3
8 = 2 since 2 · 2 · 2 = 23 = 8
Fourth
Root
√
4
number
indicates the fourth root of the number under the radical sign.
√
4
81 = 3 since 3 · 3 · 3 · 3 =
34 = 81
√
5
In an expression such as 32
Radical Sign
√
is called the radical sign.
Index
5 is called the index. (The index describes the indicated root.)
Radicand
32 is called the radicand.
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Radical
√
5
32 is called a radical (or radical expression).
5.1 Sample Set B
Find each root.
Example
5
√
25 To determine the square root of 25, we ask, "What whole number squared equals 25?" From
our √experience with multiplication, we know this number to be 5. Thus,
25 = 5
2
Check: 5 · 5 = 5 = 25
Example
6
√
32 To determine the fth root of 32, we ask, "What whole number raised to the fth power
equals
32?" This number is 2.
√
5
32 = 2
5
Check: 2 · 2 · 2 · 2 · 2 = 2 = 32
5
5.2 Practice Set B
Find the following roots using only a knowledge of multiplication.
Exercise
7
√
(Solution on p. 10.)
Exercise
8
√
(Solution on p. 10.)
Exercise
9
√
(Solution on p. 10.)
Exercise
10
√
(Solution on p. 10.)
64
100
3
6
64
64
6 Calculators
Calculators with the
√
x, y x , and 1/x keys can be used to nd or approximate roots.
6.1 Sample Set C
Example 7
Use the calculator to nd
√
121
Display Reads
Type
121
Press
x
√
121
11
Table 1
Example
√ 8
Find
7
2187.
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Display Reads
Type
2187
2187
Press
yx
2187
Type
7
7
Press
1/x
.14285714
Press
=
3
Table 2
√
7
2187 = 3 (Which means that 37 = 2187 .)
6.2 Practice Set C
Use a calculator to nd the following roots.
Exercise
11
√
(Solution on p. 10.)
Exercise
12
√
(Solution on p. 10.)
Exercise
13
√
(Solution on p. 10.)
Exercise
14
√
(Solution on p. 10.)
3
4
729
8503056
53361
12
16777216
7 Exercises
For the following problems, write the expressions using exponential notation.
Exercise 15
(Solution on p. 10.)
4·4
Exercise 16
12 · 12
Exercise 17
(Solution on p. 10.)
9·9·9·9
Exercise 18
10 · 10 · 10 · 10 · 10 · 10
Exercise 19
826 · 826 · 826
(Solution on p. 10.)
Exercise 20
3, 021 · 3, 021 · 3, 021 · 3, 021 · 3, 021
Exercise 21
|6 · 6 ·{z· · · · 6}
85 factors of 6
Exercise 22
2
| · 2 ·{z· · · · 2}
112 factors of 2
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(Solution on p. 10.)
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Exercise 23
6
(Solution on p. 10.)
|1 · 1 ·{z· · · · 1}
3,008 factors of 1
For the following problems, expand the terms. (Do not nd the actual value.)
Exercise 24
53
Exercise 25
(Solution on p. 10.)
74
Exercise 26
152
Exercise 27
1175
(Solution on p. 10.)
Exercise 28
616
Exercise 29
(Solution on p. 10.)
302
For the following problems, determine the value of each of the powers. Use a calculator to check each result.
Exercise 30
32
Exercise 31
(Solution on p. 10.)
42
Exercise 32
12
Exercise 33
102
(Solution on p. 10.)
Exercise 34
112
Exercise 35
122
(Solution on p. 10.)
Exercise 36
132
Exercise 37
152
(Solution on p. 11.)
Exercise 38
14
Exercise 39
(Solution on p. 11.)
34
Exercise 40
73
Exercise 41
103
(Solution on p. 11.)
Exercise 42
1002
Exercise 43
83
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(Solution on p. 11.)
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Exercise 44
55
Exercise 45
(Solution on p. 11.)
93
Exercise 46
62
Exercise 47
(Solution on p. 11.)
71
Exercise 48
128
Exercise 49
(Solution on p. 11.)
27
Exercise 50
05
Exercise 51
(Solution on p. 11.)
84
Exercise 52
58
Exercise 53
(Solution on p. 11.)
69
Exercise 54
253
Exercise 55
422
(Solution on p. 11.)
Exercise 56
313
Exercise 57
155
(Solution on p. 11.)
Exercise 58
220
Exercise 59
(Solution on p. 11.)
8162
For the following problems, nd the roots (using your knowledge of multiplication). Use a calculator to check
each result.
Exercise
60
√
9
Exercise
61
√
16
(Solution on p. 11.)
Exercise
62
√
36
Exercise
63
√
64
Exercise
64
√
121
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(Solution on p. 11.)
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Exercise
65
√
(Solution on p. 11.)
144
Exercise
66
√
169
Exercise
67
√
(Solution on p. 11.)
225
Exercise
68
√
27
3
Exercise
69
√
(Solution on p. 11.)
32
5
Exercise
70
√
256
4
Exercise
71
√
(Solution on p. 11.)
216
3
Exercise
72
√
7
1
Exercise
73
√
(Solution on p. 11.)
400
Exercise
74
√
900
Exercise 75
√
(Solution on p. 11.)
10, 000
Exercise
76
√
324
Exercise 77
√
3, 600
For the following problems, use a calculator with the keys
(Solution on p. 11.)
√
x, y x , and 1/x to nd each of the values.
Exercise
78
√
676
Exercise 79
√
1, 156
(Solution on p. 11.)
Exercise 80
√
46, 225
Exercise 81
√
17, 288, 964
(Solution on p. 11.)
Exercise 82
√
3
3, 375
Exercise 83
√
4
331, 776
(Solution on p. 11.)
Exercise 84
√
8
5, 764, 801
Exercise 85
√
12
16, 777, 216
Exercise 86
√
8
16, 777, 216
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(Solution on p. 11.)
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Exercise 87
√
10
9, 765, 625
9
(Solution on p. 12.)
Exercise 88
√
4
160, 000
Exercise 89
√
3
531, 441
(Solution on p. 12.)
7.1 Exercises for Review
Exercise 90
() Use the numbers 3, 8, and 9 to illustrate the associative property of addition.
Exercise 91
(Solution on p. 12.)
() In the multiplication 8 · 4 = 32, specify the name given to the numbers 8 and 4.
Exercise 92
() Does the quotient 15 ÷ 0 exist? If so, what is it?
Exercise 93
() Does the quotient 0 ÷ 15exist? If so, what is it?
Exercise 94
(Solution on p. 12.)
() Use the numbers 4 and 7 to illustrate the commutative property of multiplication.
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Solutions to Exercises in this Module
Solution to Exercise (p. 2)
372
Solution to Exercise (p. 2)
165
Solution to Exercise (p. 2)
910
Solution to Exercise (p. 2)
85 · 85 · 85
Solution to Exercise (p. 2)
4·4·4·4·4·4·4
Solution to Exercise (p. 2)
1, 739 · 1, 739
Solution to Exercise (p. 4)
8
Solution to Exercise (p. 4)
10
Solution to Exercise (p. 4)
4
Solution to Exercise (p. 4)
2
Solution to Exercise (p. 5)
9
Solution to Exercise (p. 5)
54
Solution to Exercise (p. 5)
231
Solution to Exercise (p. 5)
4
Solution to Exercise (p. 5)
42
Solution to Exercise (p. 5)
94
Solution to Exercise (p. 5)
8263
Solution to Exercise (p. 5)
685
Solution to Exercise (p. 6)
13008
Solution to Exercise (p. 6)
7·7·7·7
Solution to Exercise (p. 6)
117 · 117 · 117 · 117 · 117
Solution to Exercise (p. 6)
30 · 30
Solution to Exercise (p. 6)
4 · 4 = 16
Solution to Exercise (p. 6)
10 · 10 = 100
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Solution to Exercise (p. 6)
12 · 12 = 144
Solution to Exercise (p. 6)
15 · 15 = 225
Solution to Exercise (p. 6)
3 · 3 · 3 · 3 = 81
Solution to Exercise (p. 6)
10 · 10 · 10 = 1, 000
Solution to Exercise (p. 6)
8 · 8 · 8 = 512
Solution to Exercise (p. 7)
9 · 9 · 9 = 729
Solution to Exercise (p. 7)
71 = 7
Solution to Exercise (p. 7)
2 · 2 · 2 · 2 · 2 · 2 · 2 = 128
Solution to Exercise (p. 7)
8 · 8 · 8 · 8 = 4, 096
Solution to Exercise (p. 7)
6 · 6 · 6 · 6 · 6 · 6 · 6 · 6 · 6 = 10, 077, 696
Solution to Exercise (p. 7)
42 · 42 = 1, 764
Solution to Exercise (p. 7)
15 · 15 · 15 · 15 · 15 = 759, 375
Solution to Exercise (p. 7)
816 · 816 = 665, 856
Solution to Exercise (p. 7)
4
Solution to Exercise (p. 7)
8
Solution to Exercise (p. 8)
12
Solution to Exercise (p. 8)
15
Solution to Exercise (p. 8)
2
Solution to Exercise (p. 8)
6
Solution to Exercise (p. 8)
20
Solution to Exercise (p. 8)
100
Solution to Exercise (p. 8)
60
Solution to Exercise (p. 8)
34
Solution to Exercise (p. 8)
4,158
Solution to Exercise (p. 8)
24
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Solution to Exercise (p. 8)
4
Solution to Exercise (p. 9)
5
Solution to Exercise (p. 9)
81
Solution to Exercise (p. 9)
8 is the multiplier; 4 is the multiplicand
Solution to Exercise (p. 9)
Yes; 0
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