Download Roots and Radicals

9.1
Roots and Radicals
9.1
OBJECTIVES
1. Use the radical notation to represent roots
2. Distinguish between rational and irrational
numbers
In Chapter 3, we discussed the properties of exponents. Over the next four sections, we will
work 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
NOTE The symbol 1
first
appeared in print in 1525. In
Latin, “radix” means root, and
this was contracted to a small r.
The present symbol may have
evolved from the manuscript
form of that small r.
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 two
square roots. In general, if x2 a, we call x a square root of a.
We are now ready for our new notation. The symbol 1 is called a radical sign. We
saw above that 3 was the positive square root of 9. We also call 3 the principal square root
of 9 and can write
19 3
to indicate that 3 is the principal square root of 9.
Definitions: Square Root
1a is the positive (or principal) square root of a. It is the positive number whose
square is a.
Example 1
Finding Principal Square Roots
Find the following square roots.
(a) 149 7
(b)
4
2
A9
3
Because 7 is the positive number we must square to get 49.
Because
2
4
is the positive number we must square to get .
3
9
© 2001 McGraw-Hill Companies
CHECK YOURSELF 1
Find the following square roots.
(a) 164
NOTE When you use the
radical sign, you are referring
to the positive square root:
125 5
(b) 1144
(c)
16
A 25
Each positive number has two square roots. For instance, 25 has square roots of 5 and
5 because
52 25
and
(5)2 25
695
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CHAPTER 9
EXPONENTS AND RADICALS
If you want to indicate the negative square root, you must use a minus sign in front of the
radical.
125 5
Example 2
Finding Square Roots
Find the following square roots.
(a) 1100 10
The principal root
(b) 1100 10
The negative square root
9
3
(c) 4
A 16
CHECK YOURSELF 2
Find the following square roots.
(a) 116
Be Careful! Do not confuse
19
with
19
The expression 19 is 3,
whereas 19 is not a real
number.
Every number that we have encountered in this text is a real number. The square roots of
negative numbers are not real numbers. For instance, 19 is not a real number because
there is no real number x such that
x2 9
Example 3 summarizes our discussion thus far.
Example 3
Finding Square Roots
Evaluate each of the following square roots.
(a) 136 6
(b) 1121 11
(c) 164 8
(d) 164 is not a real number.
(e) 10 0
(Because 0 0 0)
CHECK YOURSELF 3
Evaluate, if possible.
(a) 181
(b) 149
(c) 149
(d) 149
© 2001 McGraw-Hill Companies
C A U TI ON
16
(c) A 25
(b) 116
ROOTS AND RADICALS
SECTION 9.1
697
All calculators have square root keys, but the only integers for which the calculator gives
the 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 use
your calculator to approximate square roots.
Example 4
Approximating Square Roots
Use your calculator to approximate each square root to the nearest hundredth.
NOTE The sign means “is
approximately equal to.”
(a) 145 6.708203932 6.71
(b) 18 2.83
(c) 120 4.47
(d) 1273 16.52
CHECK YOURSELF 4
Use your calculator to approximate each square root to the nearest hundredth.
(a) 13
(b) 114
(c) 191
(d) 1756
As we mentioned earlier, finding the square root of a number is the reverse of squaring
a number. We can extend that idea to work with other roots of numbers. For instance, the
cube root of a number is the number we must cube (or raise to the third power) to get that
number. For example, the cube root of 8 is 2 because 23 8, and we write
3
NOTE 18 is read “the cube
3
18 2
root of 8.”
The parts of a radical expression are summarized as follows.
Definitions: Parts of a Radical Expression
3
NOTE The index for 1a is 3.
Every radical expression contains three parts as shown below. The principal nth
root of a is written as
Index
NOTE The index of 2 for
© 2001 McGraw-Hill Companies
square roots is generally not
written. We understand that
1a is the principal square root
of a.
n
1a
Radical
sign
Radicand
To illustrate, the cube root of 64 is written
Index
of 3
3
164 4
because 43 = 64. And
Index
of 4
4
181 3
is the fourth root of 81 because 34 81.
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CHAPTER 9
EXPONENTS AND RADICALS
We can find roots of negative numbers as long as the index is odd (3, 5, etc.). For
example,
3
164 4
because (4)3 64.
If the index is even (2, 4, etc.), roots of negative numbers are not real numbers. For
example,
4
116
NOTE The even power of a
real number is always positive
or zero.
is not a real number because there is no real number x such that x4 16.
The following table shows the most common roots.
Square Roots
NOTE It would be helpful for
your work here and in future
mathematics classes to
memorize these roots.
11
14
19
116
125
1
2
3
4
5
136 6
Cube Roots
149
164
181
1100
1121
7
8
9
10
11
3
11
3
18
3
127
3
164
3
1125
1
2
3
4
5
Fourth Roots
4
11
116
4
181
4
1256
4
1625
4
1
2
3
4
5
1144 12
You can use the table in Example 5, which summarizes the discussion so far.
Example 5
Evaluating Cube Roots and Fourth Roots
Evaluate each of the following.
NOTE The cube root of a
negative number will be
negative.
NOTE The fourth root of a
negative number is not a real
number.
5
(a) 132 2 because 25 32.
3
(b) 1125 5 because (5)3 125.
4
(c) 181 is not a real number.
CHECK YOURSELF 5
Evaluate, if possible.
4
(b) 116
4
(c) 1256
3
(d) 18
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 denominator
are integers and whose denominator is nonzero. The form of a rational number is
a
b
NOTE Notice that each
radicand is a perfect-square
integer (that is, an integer that
is the square of another
integer).
a and b are integers, b 0
Certain square roots are rational numbers also. For example,
14
125
and
164
represent the rational numbers 2, 5, and 8, respectively.
© 2001 McGraw-Hill Companies
3
(a) 164
ROOTS AND RADICALS
NOTE The fact that the square
root of 2 is irrational will be
proved in later mathematics
courses and was known to
Greek mathematicians over
2000 years ago.
SECTION 9.1
699
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 an
irrational number. Because the radicands are not perfect squares, the expressions 12, 13,
and 15 represent irrational numbers.
Example 6
Identifying Rational Numbers
Which of the following numbers are rational and which are irrational?
2
A3
4
A9
17
116
125
2
are irrational numbers. And 116 and 125 are rational numbers because
A3
4
2
4
.
16 and 25 are perfect squares. Also
is rational because
A9
3
A9
Here 17 and
CHECK YOURSELF 6
Which of the following numbers are rational and which are irrational?
(a) 126
NOTE The decimal
representation of a rational
number always terminates or
repeats. For instance,
3
0.375
8
5
0.454545. . .
11
NOTE 1.414 is an
approximation to the number
whose square is 2.
© 2001 McGraw-Hill Companies
(c)
6
A7
(d) 1105
(e)
16
A9
An important fact about the irrational numbers is that their decimal representations are
always nonterminating and nonrepeating. We can therefore only approximate irrational
numbers 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. For
instance, 12 is approximately 1.414 (to three decimal places), and we can write
12 1.414
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 set
of real numbers. The real numbers will represent every point that can be pictured on the
number line. Some examples are shown below.
0
NOTE For this reason we refer
to the number line as the real
number line.
(b) 149
3
34
2
15
8
10 4
The following diagram summarizes the relationships among the various numeric sets.
Real numbers
Rational numbers
Fractions
Irrational numbers
Integers
Negative
integers
Zero
Natural
numbers
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CHAPTER 9
EXPONENTS AND RADICALS
We conclude our work in this section by developing a general result that we will need
later. Let’s start by looking at two numerical examples.
222 14 2
(1)
2(2) 14 2
because (2) 4
2
2
(2)
Consider the value of 2x2 when x is positive or negative.
NOTE This is because the
In (1) when x 2:
In (2) when x 2:
principal square root of a
number is always positive or
zero.
222 2
2(2)2 2
2(2)2 (2) 2
Comparing the results of (1) and (2), we see that 2x2 is x if x is positive (or 0) and 2x2 is
x if x is negative. We can write
2x2 x
x
when x 0
when x 0
From your earlier work with absolute values you will remember that
x x
x
when x 0
when x 0
and we can summarize the discussion by writing
2x2 x
for any real number x
Example 7
Evaluating Radical Expressions
could write
2( 4)2 216 4
Evaluate each of the following.
(b) 2(4)2 4 4
(a) 252 5
CHECK YOURSELF 7
Evaluate.
(b) 2(6)2
(a) 262
CHECK YOURSELF ANSWERS
4
4
2. (a) 4; (b) 4; (c) 3. (a) 9; (b) 7; (c) 7;
5
5
(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 149 7); (c) irrational; (d) irrational
16
4
(e) because
7. (a) 6; (b) 6
A9
3
1. (a) 8; (b) 12; (c)
© 2001 McGraw-Hill Companies
NOTE Alternatively in (b), we
Name
Exercises
9.1
Section
Date
Evaluate, if possible.
ANSWERS
1. 116
2. 1121
1.
2.
3. 1400
4. 164
3.
4.
5. 1100
6. 1100
5.
6.
7. 181
8. 181
7.
8.
9.
16
A9
1
A 25
10. 9.
10.
11.
4
A 5
12.
4
A 25
11.
12.
3
13. 127
4
14. 181
13.
14.
3
15. 127
4
16. 116
15.
16.
4
17. 181
3
18. 164
17.
18.
19.
© 2001 McGraw-Hill Companies
3
19. 127
3
20. 18
20.
21.
4
21. 1625
3
22. 11000
22.
23.
1
23.
A 27
3
24.
8
24.
A 27
3
701
ANSWERS
25.
Which of the following roots are rational numbers and which are irrational numbers?
26.
25. 119
26. 136
27. 1100
28. 17
3
30. 18
27.
28.
29.
30.
29. 19
31.
3
32.
33.
4
31. 116
32.
4
A9
34.
35.
33.
36.
4
A7
3
34. 15
37.
38.
3
35. 127
4
36. 181
39.
40.
Use your calculator to approximate the square root to the nearest hundredth.
41.
42.
37. 111
38. 114
39. 17
40. 123
41. 146
42. 178
43.
44.
45.
702
43.
2
A5
44.
3
A4
45.
8
A9
46.
7
A 15
© 2001 McGraw-Hill Companies
46.
ANSWERS
47. 118
47.
48. 131
48.
49. 127
49.
50. 165
50.
51.
For exercises 51 to 56, find the two expressions that are equivalent.
52.
51. 116, 116, 4
52. 125, 5, 125
53.
54.
3
3
53. 1125, 1125, 5 5
5
54. 132, 132, 2 55.
56.
4
3
55. 110,000, 100, 11000
3
2
56. 10 , 110,000, 1100,000
57.
58.
In exercises 57 to 62, label the statement as true or false.
59.
60.
16
4
57. 216x 4x
2
58. 2(x 4) x 4
61.
62.
4 4
59. 216x y is a real number
2
2
60. 2x y x y
63.
64.
65.
2x2 25
1x 5
61.
x5
62. 12 16 18
66.
63. Dimensions of a square. The area of a square is 32 square feet (ft2). Find the length
© 2001 McGraw-Hill Companies
of 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 to
the nearest hundredth.
65. Radius of a circle. The area of a circle is 147 ft2. Find the radius to the nearest
hundredth.
66. Radius of a circle. If the area of a circle is 72 square centimeters (cm2), find the
radius to the nearest hundredth.
703
ANSWERS
67.
67. Freely falling objects. The time in seconds (s) that it takes for an object to fall from
1
1s, in which s is the distance fallen. Find the time required for
4
an object to fall to the ground from a building that is 800 ft high.
68.
rest is given by t 69.
70.
68. Freely falling objects. Find the time required for an object to fall to the ground from
71.
a building that is 1400 ft high. (Use the formula in exercise 67.)
72.
In exercises 69 to 71, the area is given in square feet. Find the length of a side of the
square. Round your answer to the nearest hundredth of a foot.
73.
74.
75.
69.
70.
71.
2
2
10 ft
13 ft
2
17 ft
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, of
a binomial and the opposite of a binomial. To illustrate, use 4 17.
74. Determine two consecutive integers whose square roots are also consecutive
75. Determine the missing binomial in the following: (13 2)(
704
) 1.
© 2001 McGraw-Hill Companies
integers.
ANSWERS
76. Try the following using your calculator.
76.
(a) Choose a number greater than 1 and find its square root. Then find the square root
of 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 find
the square root of the result, and continue in this manner, observing successive
square roots. Do these numbers seem to be approaching a certain value? If so,
what?
77.
78.
77. (a) Can a number be equal to its own square root?
79.
(a)
(b)
(c)
(d)
(e)
a.
(b) Other than the number(s) found in part a, is a number always greater than its
square root? Investigate.
b.
78. Let a and b be positive numbers. If a is greater than b, is it always true that the square
c.
root of a is greater than the square root of b? Investigate.
d.
e.
79. Suppose that a weight is attached to a string of length L, and the other end of the
string is held fixed. If we pull the weight and then release it, allowing the weight to
swing 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 and
then back. The following formula may be used to describe the relationship between T
and L.
f.
g.
h.
L
T 2p
Ag
If L is expressed in centimeters, then g 980 cm/s2. For each of the following string
lengths, calculate the corresponding period. Round to the nearest tenth of a second.
© 2001 McGraw-Hill Companies
(a) 30 cm
(b) 50 cm
(c) 70 cm
(d) 90 cm
(e) 110 cm
Getting Ready for Section 9.2 [Section 1.7]
Find each of the following products.
(a) (4x2)(2x)
(e) (27p6)(3p)
(b) (9a4)(5a)
(f) (81s4)(s3)
(c) (16m2)(3m)
(g) (100y4)(2y)
(d) (8b3)(2b)
(h) (49m6)(2m)
705
Answers
1. 4
3. 20
5. 10
15. 3
1
19. 3
21. 5
23.
25. Irrational
3
31. Rational
33. Irrational
35. Rational
41. 6.78
43. 0.63
45. 0.94
47. 4.24
11. Not a real number
51.
57.
67.
73.
13. 3
3
4
3
17. Not a real number
7. Not a real number
3
9.
27. Rational
4
37. 3.32
49. 5.20
29. Irrational
39. 2.65
3
53. 1125, 1125
55. 110,000, 11000
116, 4
False
59. True
61. False
63. 5.66 ft
65. 6.84 ft
7.07 s
69. 3.16 ft
71. 4.12 ft
Conjugate: 4 17; opposite: 4 17
75. 13 2
77.
a. 8x3
g. 200y5
b. 45a5
h. 98m7
© 2001 McGraw-Hill Companies
79. (a) 1.1 s; (b) 1.4 s; (c) 1.7 s; (d) 1.9 s; (e) 2.1 s
c. 48m3
d. 16b4
e. 81p7
f. 81s7
706