# Finding and Estimating Square Roots

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```					            Finding and Estimating
Square Roots
ALGEBRA 1 LESSON 10-3

(For help, go to Lessons 1-2 and 8-5.)

Simplify each expression.
1. 112           2. (–12)2      3. –(12)2              4. 1.52

5. 0.62          6. 1    2      7.    2   2            8. 4     2

2
–3                     5

10-3
Finding and Estimating
Square Roots
ALGEBRA 1 LESSON 10-3

Solutions
1. 112 = 11 • 11 = 121                    2. (–12)2 = (–12)(–12) = 144

3. (–12)2 = –(12)(12) = –144           4. 1.52 = (1.5)(1.5) = 2.25

2
5. 0.62 = (0.6)(0.6) = 0.36            6. 1        = 1• 1 = 1
2         2   2   4
2                                2
7. – 2            = –2    2
–3   = 4      8. 4        = 4 • 4 = 16
3               3          9          5         5   5       25

10-3
10.3: Finding and estimating
Vocabulary: square roots
Square root: A number a is a square root of b if a2 = b

Radicand: The number inside of the radical sign.

The index gives the degree of the root. It is
Index:
the small number in the “hook” of the radical
sign.
Principal Root: When a number has two roots, the principle
root is the positive root.
Property of Square Roots:

a a
2
Finding and Estimating
Square Roots
ALGEBRA 1 LESSON 10-3

Simplify each expression.

a.     25 = 5                positive square root

9     3                                    3      3
b. ±        = ±5             The square roots are     and – .
25                                         5      5

c. –     64 = –8             negative square root

d.     –49 is undefined      For real numbers, the square root of a negative
number is undefined.

e. ±      0= 0               There is only one square root of 0.

10-3
Rational:                              Irrational:
A number that can be                   A number that canNOT be
written in RATIO form.                 written as a RATIO.

Tell whether each expression is rational or irrational.

a. ±    144 = ± 12                rational

1
b. –    5
= –0.44721359 . . .     irrational

c. –    6.25 = –2.5               rational

1
d.       = 0.3                    rational
9

e.     7 = 2.64575131 . . .       irrational

10-3
Table of Perfect Squares: page 750

N   N2             N        N2
1   1             9         81
2   4             10        100
3   9             11        121
4   16            13        169
5   25            14        196
6   36            15        225
7   49            16        256
8   64            17        289
N      N2
1      1
2      4
3      9
4      16
5      25
6      36
Between what two consecutive integers is   28.34 ?   7      49
8      64
28.34 is between the two consecutive
25 <     28.34 <   36
perfect squares 25 and 36.

The square roots of 25 and 36 are 5 and
5<     28.34 < 6
6, respectively.

28.34 is between 5 and 6.
10-3
Find   28.34 to the nearest hundredth.

28.34   5.323532662        Use a calculator.

5.32               Round to the nearest hundredth.

10-3
Suppose a rectangular field has a length three times its width
x. The formula d = x2 + (3x)2 gives the distance of the diagonal of a
rectangle. Find the distance of the diagonal across the field if x = 8 ft.

d=    x2 + (3x)2

d=    82 + (3 • 8)2         Substitute 8 for x.

d=    64 + 576               Simplify.

d=    640

d    25.3                   Use a calculator. Round to the nearest tenth.

The diagonal is about 25.3 ft long.

10-3
Finding and Estimating
Square Roots
ALGEBRA 1 LESSON 10-3

1. Simplify each expression.
4      2
a.       196 14        b. ±             ±5
25

2. Tell whether each expression is rational, irrational, or undefined.
a. ±       3           b.     –25                   c. –   2.25
5
irrational             undefined                 rational
3. Between what two consecutive integers is –      54?
–8 and –7
4. The formula s = 13.5d estimates the speed s in miles per hour that a
car was traveling, when it applied its brakes and left a skid mark d feet
long on a wet road. Estimate the speed of a car that left a 120 foot
long skid mark.

10-3
Finding and Estimating
Square Roots
ALGEBRA 1 LESSON 10-3

pages 526–528 Exercises   12. ± 0.1         24. –12.25
1. 13                 13. irrational    25. 0.93
2. 20                 14. rational      26. ± 20
3. 1                  15. irrational    27. 0
3
4. 30                 16. rational      28. ± 25
5. 0.5                17. 5 and 6       29. ±3
7
6. 6                  18. 5 and 6       30. ± 1.3
7
7. –1.1               19. –12 and –11   31. ±1
9
8. 1.4                20. 13 and 14     32. ± 27
9. 0.6                21. 3.46          33. ± 1.5
10. –12               22. –14.25        34. ± 16
11. 5                 23. 107.47        35. ± 0.1
4

10-3
Finding and Estimating
Square Roots
ALGEBRA 1 LESSON 10-3

36. ± 8                 47. 6.40                    54. false;   1=1
11
37. ± 202               48. 8.66                    55. true

38. 1            49. Answers may vary. Sample:     56. true
The first expression means    57. False; answers may vary.
39.   a. 8450 km     the neg. square root of 1 and
b. 7684 km                                       Sample: 4 + 9 = 4 + 9.
/
the second expression means
40.   21             the pos. square root of 1.    58. False; answers may vary.
50. Answers may vary.                 Sample: 12 and 3 are
41.   – 2
irrational but 36 is rational.
5            Sample: 3 and 4
42.   1.41                                         59. a. 4 units2
51. 4
43.   1.26       52. a. 5 s                            b. 1 unit2
2
44.   –5.48          b. 10 s                           c. 2 units2
c. No; the object takes
45.   –33               twice as long to fall.         d. 2 units
46.   –0.8       53. False; zero has one square root.

10-3
Finding and Estimating
Square Roots
ALGEBRA 1 LESSON 10-3

60. 1.3                 67.          70.
61. 6
62. 2
9
63. 9
64. 11                  68.          71.
65. 128
66.

69.          72. d 2 – 81
73. 9t 2 – 25
74. 8075

10-3
Finding and Estimating
Square Roots
ALGEBRA 1 LESSON 10-3

75. x2 + 26x + 169
76. 16y2 – 56y + 49
77. 10,201
78. 813,604
79. 36k2 – 49
80. 144b2 – 168b + 49
81. –2
82. 3
16
83. – 3
4
84. –972
85. 1
12
86. 81
4

10-3

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