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Roots and Radicals9.1
9.1 OBJECTIVES
1. Use the radical notation to represent roots2. Distinguish between rational and irrational
numbers
In Chapter 3, we discussed the properties of exponents. Over the next four sections, we willwork with a new notation that “reverses” the process of raising to a power.
From our work in Chapter 0, we know that when we have a statement such as
x2 � 9
it is read as “x squared equals 9.”Here we are concerned with the relationship between the variable x and the number 9. We
call that relationship the square root and say, equivalently, that “x is the square root of 9.”We know from experience that x must be 3 (because 32 � 9) or �3 [because (�3)2 � 9].
We see that 9 has two square roots, 3 and �3. In fact, every positive number will have twosquare roots. In general, if x2 � a, we call x a square root of a.
We are now ready for our new notation. The symbol is called a radical sign. Wesaw above that 3 was the positive square root of 9. We also call 3 the principal square rootof 9 and can write
to indicate that 3 is the principal square root of 9.
19 � 3
1
695
is the positive (or principal) square root of a. It is the positive number whosesquare is a.1a
Definitions: Square Root
Finding Principal Square Roots
Find the following square roots.
(a) � 7 Because 7 is the positive number we must square to get 49.
(b) Because is the positive number we must square to get .49
23
�2
3A4
9
149
Example 1
NOTE The symbol firstappeared in print in 1525. InLatin, “radix” means root, andthis was contracted to a small r.The present symbol may haveevolved from the manuscriptform of that small r.
1
Each positive number has two square roots. For instance, 25 has square roots of 5 and�5 because
52 � 25 and (�5)2 � 25
Find the following square roots.
(a) (b) (c) A16
251144164
C H E C K Y O U R S E L F 1
NOTE When you use theradical sign, you are referringto the positive square root:125 � 5
696 CHAPTER 9 EXPONENTS AND RADICALS
If you want to indicate the negative square root, you must use a minus sign in front of theradical.
�125 � �5
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Finding Square Roots
Find the following square roots.
(a) � 10 The principal root
(b) � �10 The negative square root
(c) � �3
4�A 9
16
�1100
1100
Example 2
Find the following square roots.
(a) (b) (c) �A16
25�116116
C H E C K Y O U R S E L F 2
Every number that we have encountered in this text is a real number. The square roots ofnegative numbers are not real numbers. For instance, is not a real number becausethere is no real number x such that
x2 � �9
Example 3 summarizes our discussion thus far.
1�9C A U T I O N
Be Careful! Do not confuse
with
The expression is �3,whereas is not a realnumber.
1�9�19
1�9�19
Example 3
Finding Square Roots
Evaluate each of the following square roots.
(a) � 6 (b) � 11
(c) � �8 (d) is not a real number.
(e) � 0 (Because 0 � 0 � 0)10
1�64�164
1121136
Evaluate, if possible.
(a) (b) (c) (d) 1�49�149149181
C H E C K Y O U R S E L F 3
ROOTS AND RADICALS SECTION 9.1 697
All calculators have square root keys, but the only integers for which the calculator givesthe exact value of the square root are perfect square integers. For all other positive integers,a calculator gives only an approximation of the correct answer. In Example 4 you will useyour calculator to approximate square roots.
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Use your calculator to approximate each square root to the nearest hundredth.
(a) (b) (c) (d) 175619111413
C H E C K Y O U R S E L F 4
As we mentioned earlier, finding the square root of a number is the reverse of squaringa number. We can extend that idea to work with other roots of numbers. For instance, thecube root of a number is the number we must cube (or raise to the third power) to get thatnumber. For example, the cube root of 8 is 2 because 23 � 8, and we write
The parts of a radical expression are summarized as follows.
13 8 � 2
NOTE The � sign means “isapproximately equal to.”
NOTE is read “the cuberoot of 8.”
13 8
Approximating Square Roots
Use your calculator to approximate each square root to the nearest hundredth.
(a) (b)
(c) (d) � 16.521273� 4.47120
� 2.8318� 6.708203932 � 6.71145
Example 4
Every radical expression contains three parts as shown below. The principal nthroot of a is written as
Index
Radical Radicandsign
1n a
Definitions: Parts of a Radical Expression
To illustrate, the cube root of 64 is written
Indexof 3
because 43 = 64. And
Indexof 4
is the fourth root of 81 because 34 � 81.
14 81 � 3
13 64 � 4
NOTE The index for is 3.13 a
NOTE The index of 2 forsquare roots is generally notwritten. We understand that
is the principal square rootof a.1a
The radical notation helps us to distinguish between two important types of numbers:rational numbers and irrational numbers.
A rational number can be represented by a fraction whose numerator and denominatorare integers and whose denominator is nonzero. The form of a rational number is
a and b are integers, b � 0
Certain square roots are rational numbers also. For example,
represent the rational numbers 2, 5, and 8, respectively.
14 125 and 164
a
b
698 CHAPTER 9 EXPONENTS AND RADICALS
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NOTE The even power of areal number is always positiveor zero.
NOTE It would be helpful foryour work here and in futuremathematics classes tomemorize these roots.
Square Roots Cube Roots Fourth Roots
1144 � 12136 � 6
14 625 � 513 125 � 51121 � 11125 � 5
14 256 � 413 64 � 41100 � 10116 � 4
14 81 � 313 27 � 3181 � 919 � 3
14 16 � 213 8 � 2164 � 814 � 2
14 1 � 113 1 � 1149 � 711 � 1
You can use the table in Example 5, which summarizes the discussion so far.
Evaluating Cube Roots and Fourth Roots
Evaluate each of the following.
(a) � 2 because 25 � 32.
(b) � �5 because (�5)3 � �125.
(c) is not a real number.14 �81
13 �125
15 32
Example 5
Evaluate, if possible.
(a) (b) (c) (d) 13 �814 �25614 1613 64
C H E C K Y O U R S E L F 5
NOTE The cube root of anegative number will benegative.
NOTE The fourth root of anegative number is not a realnumber.
NOTE Notice that eachradicand is a perfect-squareinteger (that is, an integer thatis the square of anotherinteger).
We can find roots of negative numbers as long as the index is odd (3, 5, etc.). Forexample,
because (�4)3 � �64.If the index is even (2, 4, etc.), roots of negative numbers are not real numbers. For
example,
is not a real number because there is no real number x such that x4 � �16.The following table shows the most common roots.
14 �16
13 �64 � �4
ROOTS AND RADICALS SECTION 9.1 699©
200
1 M
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NOTE The fact that the squareroot of 2 is irrational will beproved in later mathematicscourses and was known toGreek mathematicians over2000 years ago.
Example 6
Identifying Rational Numbers
Which of the following numbers are rational and which are irrational?
Here and are irrational numbers. And and are rational numbers because
16 and 25 are perfect squares. Also is rational because A4
9�
2
3.A4
9
125116A2
317
A2
3 A4
9 17 116 125
An irrational number is a number that cannot be written as the ratio of two integers.For example, the square root of any positive number that is not itself a perfect square is anirrational number. Because the radicands are not perfect squares, the expressions and represent irrational numbers.15
12, 13,
An important fact about the irrational numbers is that their decimal representations arealways nonterminating and nonrepeating. We can therefore only approximate irrationalnumbers with a decimal that has been rounded off. A calculator can be used to find roots.However, note that the values found for the irrational roots are only approximations. Forinstance, is approximately 1.414 (to three decimal places), and we can write
With a calculator we find that
(1.414)2 � 1.999396
The set of all rational numbers and the set of all irrational numbers together form the setof real numbers. The real numbers will represent every point that can be pictured on thenumber line. Some examples are shown below.
The following diagram summarizes the relationships among the various numeric sets.
815�
433� 2 10 4
0
12 � 1.414
12
NOTE For this reason we referto the number line as the realnumber line.
Which of the following numbers are rational and which are irrational?
(a) (b) (c) (d) (e) A16
91105A6
7149126
C H E C K Y O U R S E L F 6
NOTE The decimalrepresentation of a rationalnumber always terminates orrepeats. For instance,
511
� 0.454545. . .
38
� 0.375
NOTE 1.414 is anapproximation to the numberwhose square is 2.
Real numbers
Rational numbers Irrational numbers
Fractions Integers
Negative Zero Naturalintegers numbers
700 CHAPTER 9 EXPONENTS AND RADICALS
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We conclude our work in this section by developing a general result that we will needlater. Let’s start by looking at two numerical examples.
(1)
because (�2)2 � 4 (2)
Consider the value of when x is positive or negative.
In (1) when x � 2: In (2) when x � �2:
2(�2)2 � �(�2) � 2
2(�2)2 � �2222 � 2
2x2
2(�2)2 � 14 � 2
222 � 14 � 2
NOTE This is because theprincipal square root of anumber is always positive orzero.
Comparing the results of (1) and (2), we see that is x if x is positive (or 0) and is�x if x is negative. We can write
From your earlier work with absolute values you will remember that
and we can summarize the discussion by writing
for any real number x2x2 � �x�
�x� � � x
�x
when x � 0
when x � 0
2x2 � � x
�x
when x � 0
when x � 0
2x22x2
Evaluate.
(a) (b) 2(�6)2262
C H E C K Y O U R S E L F 7
1. (a) 8; (b) 12; (c) 2. (a) 4; (b) �4; (c) 3. (a) 9; (b) 7; (c) �7;
(d) not a real number 4. (a) 1.73; (b) 3.74; (c) 9.54; (d) 27.50
5. (a) 4; (b) 2; (c) not a real number; (d) �2 6. (a) Irrational;
(b) rational (because ); (c) irrational; (d) irrational
(e) 7. (a) 6; (b) 6�because A16
9�
4
3�149 � 7
�4
5
4
5
C H E C K Y O U R S E L F A N S W E R S
Evaluating Radical Expressions
Evaluate each of the following.
(a) � 5 (b) � ��4� � 42(�4)2252
Example 7
NOTE Alternatively in (b), wecould write2( � 4)2 � 216 � 4
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Exercises
Evaluate, if possible.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24. A3 �8
27A3 1
27
13 100014 625
�13 �8�13 27
�13 6414 �81
14 �1613 �27
14 8113 27
A 4
25A�4
5
�A 1
25A16
9
�1811�81
1�100�1100
1641400
1121116
9.1
Name
Section Date
ANSWERS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23. 24.
701
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ANSWERS
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
702
Which of the following roots are rational numbers and which are irrational numbers?
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.
Use your calculator to approximate the square root to the nearest hundredth.
37. 38.
39. 40.
41. 42.
43. 44.
45. 46. A 7
15A8
9
A3
4A2
5
178146
12317
114111
�14 8113 �27
13 5A4
7
A4
914 16
13 813 9
171100
136119
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47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
703
47. 48.
49. 50.
For exercises 51 to 56, find the two expressions that are equivalent.
51. 52.
53. 54.
55. 56.
In exercises 57 to 62, label the statement as true or false.
57. 58.
59. is a real number 60.
61. 62.
63. Dimensions of a square. The area of a square is 32 square feet (ft2). Find the lengthof a side to the nearest hundredth.
64. Dimensions of a square. The area of a square is 83 ft2. Find the length of the side tothe nearest hundredth.
65. Radius of a circle. The area of a circle is 147 ft2. Find the radius to the nearesthundredth.
66. Radius of a circle. If the area of a circle is 72 square centimeters (cm2), find theradius to the nearest hundredth.
12 � 16 � 182x2 � 25
x � 5� 1x � 5
2x2 � y2 � x � y216x�4y�4
2(x � 4)2 � x � 4216x16 � 4x4
102, 110,000, 13 100,00014 10,000, 100, 13 1000
15 �32, �15 32, ��2�13 �125, �13 125, ��5�
�125, �5, 1�251�16, �116, �4
�165�127
�131�118
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ANSWERS
67.
68.
69.
70.
71.
72.
73.
74.
75.
704
67. Freely falling objects. The time in seconds (s) that it takes for an object to fall from
rest is given by in which s is the distance fallen. Find the time required for
an object to fall to the ground from a building that is 800 ft high.
68. Freely falling objects. Find the time required for an object to fall to the ground froma building that is 1400 ft high. (Use the formula in exercise 67.)
In exercises 69 to 71, the area is given in square feet. Find the length of a side of thesquare. Round your answer to the nearest hundredth of a foot.
69. 70. 71.
72. Is there any prime number whose square root is an integer? Explain your answer.
73. Explain the difference between the conjugate, in which the middle sign is changed, ofa binomial and the opposite of a binomial. To illustrate, use
74. Determine two consecutive integers whose square roots are also consecutive integers.
75. Determine the missing binomial in the following: (13 � 2)( ) � �1.
4 � 17.
17 ft2
10 ft2 13 ft
2
t �1
4 1s,
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76.
77.
78.
(a)(b)(c)(d)(e)
a.
b.
c.
d.
e.
f.
g.
h.
705
76. Try the following using your calculator.
(a) Choose a number greater than 1 and find its square root. Then find the square rootof the result and continue in this manner, observing the successive square roots.Do these numbers seem to be approaching a certain value? If so, what?
(b) Choose a number greater than 0 but less than 1 and find its square root. Then findthe square root of the result, and continue in this manner, observing successivesquare roots. Do these numbers seem to be approaching a certain value? If so,what?
77. (a) Can a number be equal to its own square root?
(b) Other than the number(s) found in part a, is a number always greater than itssquare root? Investigate.
78. Let a and b be positive numbers. If a is greater than b, is it always true that the squareroot of a is greater than the square root of b? Investigate.
79. Suppose that a weight is attached to a string of length L, and the other end of thestring is held fixed. If we pull the weight and then release it, allowing the weight toswing back and forth, we can observe the behavior of a simple pendulum. The period,T, is the time required for the weight to complete a full cycle, swinging forward andthen back. The following formula may be used to describe the relationship between Tand L.
If L is expressed in centimeters, then g � 980 cm/s2. For each of the following stringlengths, calculate the corresponding period. Round to the nearest tenth of a second.
(a) 30 cm (b) 50 cm (c) 70 cm (d) 90 cm (e) 110 cm
T � 2pAL
g
Getting Ready for Section 9.2 [Section 1.7]
Find each of the following products.
(a) (4x2)(2x) (b) (9a4)(5a) (c) (16m2)(3m) (d) (8b3)(2b)
(e) (27p6)(3p) (f) (81s4)(s3) (g) (100y4)(2y) (h) (49m6)(2m)
79.
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Answers
1. 4 3. 20 5. �10 7. Not a real number 9.
11. Not a real number 13. 3 15. �3 17. Not a real number
19. �3 21. 5 23. 25. Irrational 27. Rational 29. Irrational
31. Rational 33. Irrational 35. Rational 37. 3.32 39. 2.6541. 6.78 43. 0.63 45. 0.94 47. �4.24 49. �5.20
51. 53. 55.57. False 59. True 61. False 63. 5.66 ft 65. 6.84 ft67. 7.07 s 69. 3.16 ft 71. 4.12 ft73. Conjugate: opposite: 75. 77.
79. (a) 1.1 s; (b) 1.4 s; (c) 1.7 s; (d) 1.9 s; (e) 2.1 s a. 8x3 b. 45a5
c. 48m3 d. 16b4 e. 81p7 f. 81s7 g. 200y5 h. 98m7
13 � 2�4 �174 �17;
14 10,000, 13 100013 �125, �13 125�116, �4
1
3
4
3
