8.1
Roots and Radicals
8.1
OBJECTIVES
1. Use radical notation
2. Evaluate expressions that contain radicals
3. Use a calculator to estimate or evaluate radical
expressions
4. Simplify expressions that contain radicals
In Chapters 1 and 7 we discussed the properties of integer exponents. Over the next six sections, we will be working toward an extension of those properties. To achieve that objective,
we must develop a notation that “reverses” the power process.
A statement such as
x2 9
NOTE We will see later that a
negative number has no real
square roots.
is read as “x squared equals 9.”
In this section we are concerned with the relationship between the base x and the
number 9. Equivalently, we can say 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 the two square roots, 3 and 3. In fact, every positive number has two
square roots, one positive and one negative. In general,
If x2 a, we say x is a square root of a.
We also know that
33 27
and similarly we call 3 a cube root of 27. Here 3 is the only real number with that property.
Every real number (positive or negative) has one real cube root.
Definitions: Roots
In general, we can state that if
xn a
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then x is an nth root of a.
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
been used because it resembled
the manuscript form of that
small r.
We are now ready for new notation. The symbol 1 is called a radical sign. We saw above
that 3 was the positive square root of 9.
We call 3 the principal square root of 9, and we write
19 3
In some applications we will want to indicate the negative square root; to do so we must
write
19 3
NOTE You will see this used
later in our work with quadratic
equations in Chapter 9.
to indicate the negative root.
If both square roots need to be indicated, we can write
19 3
581
582
CHAPTER 8
RADICAL EXPRESSIONS
Every radical expression contains three parts, as shown below. The principal nth root of
a is written as
NOTE The index of 2 for
square roots is generally not
written. We understand that
Index
n
1a
1a
is the principal square root of a.
Radical sign
Radicand
Example 1
Evaluating Radical Expressions
Evaluate, if possible.
(a) 249 7
(b) 249 7
(c) 249 7
NOTE Notice that neither 72
(d) 249 is not a real number.
nor (7)2 equals 49.
Let’s examine part (d) more carefully. Suppose that for some real number x,
x 249
By our earlier definition, this means that
x 2 49
NOTE We consider imaginary
numbers in detail in Section 8.7.
which is impossible. There is no real square root for 49. We call 249 an imaginary
number.
CHECK YOURSELF 1
Evaluate, if possible.
(a) 264
(b) 264
(c) 264
(d) 264
Example 2
Evaluating Radical Expressions
Evaluate, if possible.
3
(a) 264 4
Because 43 64
3
NOTE Notice that the cube
(b) 264 4
root of a negative number is
negative.
(c) 264 4
3
Because (4)3 64
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Our next example considers cube roots.
ROOTS AND RADICALS
SECTION 8.1
583
CHECK YOURSELF 2
Evaluate.
3
3
(a) 2125
NOTE The word “indices” is
the plural of “index.”
3
(b) 2125
(c) 2125
Let’s consider radicals with other indices in the next example.
Example 3
Evaluating Radical Expressions
Evaluate, if possible.
4
(a) 281 3
Because 34 81
4
NOTE In general, an even root
(b) 281 is not a real number.
of a negative number is not
real; it is imaginary.
(c) 232 2
5
Because 25 32
5
(d) 232 2
Because (2)5 32
CHECK YOURSELF 3
Evaluate, if possible.
4
(a) 216
5
(b) 2243
4
(c) 216
5
(d) 2243
Note: All the numbers of our previous examples and exercises were chosen so that the results would be rational numbers. That is, our radicands were
Perfect squares: 1, 4, 9, 16, 25, . . .
Perfect cubes: 1, 8, 27, 64, 125, . . .
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and so on.
The square root of a number that is not a perfect square (or the cube root of a number
that is not a perfect cube) is not a rational number.
Expressions such as 12, 13, and 15 are irrational numbers. A calculator with a
square root key 2
will give decimal approximations for such numbers.
Example 4
NOTE On some calculators, the
square root is shown as the
“2nd function” or “inverse” of
x2. If that is the case, press the
2nd function key and then
the x2 key. On graphing
calculators, you press the
key, then enclose the
2
radicand in parentheses.
Estimating Radical Expressions
Using a calculator, find the decimal approximation for each of the following. Round all
answers to three decimal places.
(a) 217
Enter 17 in your calculator and press the 2
key. The display will read 4.123105626
(if your calculator displays 10 digits). If this is rounded to three decimal places, the result
is 4.123.
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CHAPTER 8
RADICAL EXPRESSIONS
(b) 228
The display should read 5.291502622. Rounded to three decimal places, the result is 5.292.
(c) 211
Enter 11 by first entering 11 and then pressing the key. Take the square root by
pressing the 2
key. The display will read ERROR. This indicates that 11 does not
have a real square root.
CHECK YOURSELF 4
Use a calculator to find the decimal approximation for each of the following. Round
each answer to three decimal places.
(a) 213
NOTE Not all scientific
calculators have this key.
(b) 238
(c) 221
To evaluate roots other than square roots by using scientific calculators, the key marked
y x can be used together with the INV key. (On some calculators, the INV key is
2nd F .)
Example 5
Estimating Radical Expressions
Using a calculator, find a decimal approximation for each of the following. Round each
answer to three decimal places.
4
(a) 212
NOTE Again, depending upon
your calculator, you may have
only an eight-digit display.
Enter 12 and press INV y x . Then enter 4 and press . The display will read
1.861209718. Rounded to three decimal places, the result is 1.861.
5
(b) 227
Enter 27 and press INV y x . Then enter 5 and press . The display will read
1.933182045. Rounded to three decimal places, the result is 1.933.
CHECK YOURSELF 5
4
(a) 235
5
(b) 229
A certain amount of caution should be exercised in dealing with principal even roots.
For example, consider the statement
2x 2 x
NOTE Because x 2,
First, let x 2 in Equation (1).
2x2 x
22 2 24 2
(1)
(2)
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Use a calculator to find the decimal approximation for each of the following. Round
each answer to three decimal places.
ROOTS AND RADICALS
NOTE Because here x 2,
2x 2 Z x
SECTION 8.1
585
Now let x 2.
2(2) 2 24 2
(3)
We see that statement (1) is not true when x is negative, but we can write
2x2 x
when x 0
x when x 0
From your earlier work with absolute values you will remember that
x x
when x 0
x when x 0
and we can summarize the discussion by writing
NOTE Statement (4) can be
2x2 x
(4)
extended to
n
2 x n x
when n is even.
Example 6
Evaluating Radical Expressions
Evaluate.
(a) 252 5
NOTE Alternately we could
(b) 2(4)2 4 4
write
2(4)2 216 4
4
(c) 224 2
4
(d) 2(3)4 3 3
CHECK YOURSELF 6
Evaluate.
(a) 262
(b) 2(6)2
4
(c) 234
4
(d) 2(3)4
Note: The case for roots with indices that are odd does not require the use of absolute
value, as illustrated in Example 6. For instance,
3
3
233 227 3
© 2001 McGraw-Hill Companies
3
3
2(3)3 227 3
and we see that
n
2 xn x
when n is odd.
To summarize, we can write
n
2 xn x
x
when n is even
when n is odd
Let’s turn now to a final example in which variables are involved in the radicand.
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CHAPTER 8
RADICAL EXPRESSIONS
Example 7
Simplifying Radical Expressions
Simplify the following.
3
(a) 2a3 a
(b) 216m2 4m
5
(c) 232x5 2x
determine the power of the
variable in our root by dividing
the power in the radicand by
the index. In Example 7d,
8 4 2.
4
(d) 2x8 x2
3
(e) 227y6 3y2
Because (x2)4 x8.
Do you see why?
CHECK YOURSELF 7
Simplify.
4
(a) 2x4
(b) 249w2
5
(c) 2a10
3
(d) 28y9
CHECK YOURSELF ANSWERS
1.
3.
4.
5.
7.
(a) 8; (b) 8; (c) 8; (d) not a real number
2. (a) 5; (b) 5; (c) 5
(a) 2; (b) 3; (c) not a real number; (d) 3
(a) 3.606; (b) 6.164; (c) not a real number
(a) 2.432; (b) 1.961
6. (a) 6; (b) 6; (c) 3; (d) 3
2
(a) x; (b) 7w; (c) a ; (d) 2y3
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NOTE Notice that we can
Name
Exercises
8.1
Section
Date
Evaluate each of the following roots where possible.
1. 149
3. 136
5. 181
2. 136
4. 181
6. 149
ANSWERS
1.
2.
3.
4.
5.
6.
7.
8.
7. 149
8. 125
9.
10.
3
9. 127
3
10. 164
11.
12.
3
11. 164
3
12. 1125
13.
14.
3
13. 1216
3
14. 127
15.
16.
4
15. 181
5
16. 132
17.
18.
5
17. 132
4
18. 181
19.
20.
4
19. 116
5
20. 1243
21.
22.
4
21. 116
5
22. 132
23.
© 2001 McGraw-Hill Companies
24.
5
23. 1243
4
24. 1625
25.
25.
4
A9
26.
9
A 25
26.
27.
27.
8
3
A 27
28.
27
3
A 64
28.
587
ANSWERS
29.
30.
31.
32.
33.
34.
35.
36.
2
29. 26
2
30. 29
31. 2(3)2
32. 2(5)2
3
34. 2(5)3
4
36. 2(2)4
33. 243
37.
38.
35. 234
3
4
39.
40.
Simplify each of the following roots.
41.
2
37. 2x
3
3
38. 2w
42.
5
43.
7
39. 2y5
40. 2z7
41. 29x2
42. 281y2
43. 2a4b6
44. 2w6z10
45. 216x4
46. 249y6
44.
45.
46.
47.
48.
49.
4
48. 2a18
4
50. 2a6b9
3
52. 227x3
5
54. 232m10n5
47. 2y20
3
50.
51.
49. 2m8n12
3
52.
51. 2125a3
53.
3
54.
53. 232x5y15
5
56.
Using a calculator, evaluate the following. Round each answer to three decimal places.
57.
55. 115
56. 129
57. 1156
58. 1213
59. 115
60. 179
58.
59.
60.
588
© 2001 McGraw-Hill Companies
55.
ANSWERS
3
61.
3
61. 183
62. 197
62.
63. 1123
5
64. 1283
5
3
5
63.
64.
65. 115
66. 129
65.
66.
Label each of the following statements as true or false.
67.
67. 216x
16
4x
4
3
6 9 3
6 9
69. 2(4x y ) 4x y
68. 236c 6c
2
4
68.
69.
4
70. 2(x 4) x 4
70.
4
2
71. 2x 16 x 4
3
8
2
72. 2x 27 x 3
71.
72.
4 4
73. 216x y is not a real number
3
6 6
74. 28x y is not a real number.
73.
74.
75. Is there any prime number whose square root is an integer? Explain your
75.
answer.
76.
76. Determine two consecutive integers whose square roots are also consecutive
integers.
77.
78.
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77. Try the following using your calculator.
(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?
78. (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 its square root? Investigate.
589
ANSWERS
79. Let a and b be positive numbers. If a is greater than b, is it always true that the square
root of a is greater than the square root of b? Investigate.
79.
80. (a)
(b)
(c)
(d)
(e)
80. 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.
81. (a)
(b)
(c)
(d)
(e)
(f)
T 2p
82. (a)
(b)
(c)
(d)
L
Ag
If L is expressed in centimeters, then g 980 cms2. For each of the following string
lengths, 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
81. In parts (a) through (f), evaluate when possible.
(a) 14 9
(b) 14 19
(c) 19 16
(d) 19 116
(e) 1(4)(25)
(f) 14 125
(g) Based on parts (a) through (f), make a general conjecture concerning 1ab. Be
careful to specify any restrictions on possible values for a and b.
82. In parts (a) through (d), evaluate when possible.
(a) 19 16
(b) 19 116
(c) 136 64
(d) 136 164
(e) Based on parts (a) through (d), what can you say about 1a b and 1a 1b?
Answers
25.
39.
51.
61.
71.
3. 6
15. 3
5. 9
17. 2
7. Not a real number
9. 3
11. 4
19. 2
21. Not a real number
23. 3
2
2
27.
29. 6
31. 3
33. 4
35. 3
37. x
3
3
y
41. 3x
43. a2b3
45. 4x2
47. y5
49. m2n3
3
5a
53. 2xy
55. 3.873
57. 12.490
59. Not a real number
4.362
63. 2.618
65. 2.466
67. False
69. True
False
73. False
75. No
77.
79.
81. (a) 6; (b) 6; (c) 12; (d) 12; (e) 10; (f) Not possible
590
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1. 7
13. 6