LESSON
Page 1 of 5
9.1
Square Roots
BEFORE
Vocabulary
square root, p. 453
perfect square, p. 454
radical expression,
p. 454
You found squares of
numbers.
Now
WHY?
You’ll find and approximate
square roots of numbers.
So you can find a person’s
running speed, as in Ex. 59.
Human Chess In September of
every even-numbered year, people
in Marostica, Italy, play an unusual
chess game. Each chess piece is
portrayed by a person. The people
portraying the knights are even on
horseback!
The chessboard is a square with
an area of 324 square meters. What
is the length of each side of the
board? To answer this question,
you need to find the square root
of 324.
A square root of a number n is a number m such that m 2 n. Every
positive number has two square roots. One square root is positive
and the other is negative. The radical sign, , represents a
nonnegative square root. The symbol , read “plus or minus,” refers
to both square roots of a positive number. For example:
100
10
Positive square root of 100
100
10
Negative square root of 100
100
10
Positive or negative square root of 100
Zero has only one square root, itself.
Example 1
Study Strategy
In Example 1, it doesn’t make
sense to find the negative
square root of 324, because
length cannot be negative.
Finding a Square Root
The chessboard described above is a square with an area of
324 square meters, so the length of each side of the chessboard is the
positive square root of 324.
324
18 because 182 324.
Answer The length of each side of the chessboard is 18 meters.
Checkpoint
Find the square roots of the number.
1. 16
2. 64
3. 144
Lesson 9.1
4. 256
Square Roots
453
Page 2 of 5
Approximating Square Roots A perfect square is a number that is the
square of an integer. For example, 1, 4, and 9 are perfect squares.
Note Worthy
You may find it helpful to make
a list of the first 20 perfect
squares and their square roots
in your notebook and
memorize them. You can find a
table of squares and square
roots on p. 822.
1 12, 4 22, and 9 32
You can use perfect squares to approximate a square root of a number.
Example 2
Approximating a Square Root
Approximate 51
to the nearest integer.
The perfect square closest to, but less than, 51 is 49. The perfect
square closest to, but greater than, 51 is 64. So, 51 is between 49 and
64. This statement can be expressed by the compound inequality
49 < 51 < 64.
49 < 51 < 64
Identify perfect squares closest to 51.
49
< 51
< 64
7 < 51
<8
Take positive square root of each number.
Evaluate square root of each perfect square.
Answer Because 51 is closer to 49 than to 64, 51
is closer to 7 than
≈ 7.
to 8. So, to the nearest integer, 51
Checkpoint
5. Approximate 125
to the nearest whole number.
Example 3
Tech Help
When you enter a square root
on the calculator, you should
also enter a right parenthesis
to close the left parenthesis
that the calculator enters.
Although the calculator shows
8 digits for the decimal part of
the square root in Example 3,
the decimal actually continues
without end.
Using a Calculator
Use a calculator to approximate 515
. Round to the nearest tenth.
Keystrokes
[] 515
(515)
22.69361144
Answer
515
≈ 22.7
Radical Expressions A radical expression is an expression that involves
a radical sign. The horizontal bar in a radical sign is a grouping symbol.
When you evaluate a radical expression, evaluate the expression inside
the radical symbol before finding the square root.
Example 4
Evaluating a Radical Expression
Evaluate 2a
b 2 when a 11 and b 5.
b 2 211
52
2a
454
Chapter 9
Substitute 11 for a and 5 for b.
236
Evaluate expression inside radical symbol.
2 p 6 12
Evaluate square root. Multiply.
Real Numbers and Right Triangles
Page 3 of 5
Checkpoint
Evaluate the expression when a 12 and b 4.
6. a
b
7.
Example 5
b2 a
8. 3ab
1
Solving an Equation Using Square Roots
Physics An amusement park ride includes a free fall drop of 272 feet.
You can use the equation d 16t 2 to determine the time t in seconds
that it takes a dropped object to fall a distance of d feet. How long
does the free fall part of the ride take?
Solution
d 16t 2
272 16t
17 t
2
2
Write original equation.
Substitute 272 for d.
Divide each side by 16.
17
t
Use definition of square root.
4.1 ≈ t
Use a calculator to approximate square root.
Answer Because only the positive solution makes sense in this
situation, the free fall part of the ride takes about 4.1 seconds.
9.1
Exercises
INTERNET
More Practice, p. 811
CLASSZONE.COM
eWorkbook Plus
Guided Practice
Vocabulary Check
1. Describe and give an example of a perfect square.
2. You know that one square root of a number x is 9. What is the other
square root? What is the value of x?
Skill Check
Find the square roots of the number.
3. 4
4. 36
5. 121
6. 225
Approximate the square root to the nearest integer.
7. 10
8. 84
9. 151
10. 200
Solve the equation.
11. a 2 9
Guided Problem Solving
12. n 2 25
13. 361 x 2
14. 400 y 2
15. Eiffel Tower The base of the Eiffel Tower is a square with an area of
15,625 square feet. What is the length of a side of the base?
1
Write an equation that relates base area A and side length s.
2
Substitute 15,625 for A in the equation in Step 1 and solve for s.
Lesson 9.1
Square Roots
455
Page 4 of 5
Practice and Problem Solving
Homework Help
Example
1
2
3
4
5
Exercises
16–24
25–32, 54–57
33–40
41–44
45–53, 58–59
Online Resources
CLASSZONE.COM
• More Examples
• eTutorial Plus
In the following exercises, you may find it helpful to use a calculator for
approximating square roots.
Find the square roots of the number.
16. 25
17. 169
18. 81
19. 289
20. 1024
23. 900
21. 484
22. 1600
24. Geometry The area of a square is 49 square feet. Find the side length.
Approximate the square root to the nearest integer.
25. 38
26. 120
27. 148
28. 17
29. 78
30. 250
31. 15.3
32. 7.4
Use a calculator to approximate the square root. Round to the
nearest tenth.
33. 3
34. 10
35. 86
36. 110
37. 33
38. 1325
39. 19.5
40. 6.92
Evaluate the expression when a 48 and b 12.
41. a
b
42. a
b
4
43. 3ab
44.
b2 (
a 15)
Solve the equation. Round to the nearest tenth if necessary.
45. x 2 49
46. y 2 676
47. 441 t 2
48. n2 576
49. 20 m 2
50. c 2 125
51. 5y 2 110
52. 200 16t 2
53. Critical Thinking Write an equation that has exactly two solutions, 1.5
and 1.5.
In Exercises 54–57, match the number with a point on the number line.
A
0
1
54. 15
B
2
3
55. 2
4
C
5
D
6
7
56. 95
8
9
10
11
57. 27
58. Photography You can use the following rule of thumb when
photographing fireworks: The f-stop, a number that describes the size
of the opening of the camera lens, should be the number closest to the
square root of the film speed. You have a camera with f-stop numbers
2.8, 4, 5.6, 8, 11, 16, and 22. Which f-stop should you use to photograph
fireworks if you are using a film speed of 64? of 100?
59. Running You can use the formula l 0.0625s2 to approximate the
maximum running speed s (in meters per second) that a person with
leg length l (in centimeters) can sustain. Find the maximum running
speed for a person with a leg length of 64 centimeters.
Solve the equation. Round to the nearest hundredth if necessary.
Fireworks near the Space
Needle in Seattle, Washington
456
Chapter 9
60. 15 2h 2 3
61. 162 0.5t 2
62. 1400 10z 2 2
63. 3x 2 5 30
64. 1.5n2 7 20
65. 2a 2 1 98
Real Numbers and Right Triangles
Page 5 of 5
66. Consider the function y x
.
a. Make a table of ordered pairs (x, y) for x 0, 1, 4, 9, 16, and 25.
b. Plot the ordered pairs from part (a) on a coordinate plane.
c. Writing Is y x a linear function? Explain.
67. Extended Problem Solving A tsunami is an ocean wave that moves
very fast in deep water, but slows as it reaches shallow water. As the
wave slows, it rises to great heights, often causing enormous
destruction on land. A tsunami’s speed s (in feet per second) and the
depth d of the water (in feet) are related by the equation s 2 32d.
Suppose an earthquake at sea produces a tsunami in water
15,000 feet deep.
In the
Real World
Tsunamis An earthquake
occurred off the coast of Chile
in 1960, generating a tsunami.
The tsunami reached Japan,
about 10,000 miles away,
22 hours later. To the nearest
mile per hour, how fast did the
tsunami travel?
a. Calculate Find the original speed of the wave to the nearest mile
per hour.
b. Apply The wave enters a harbor 45 feet deep. Find the change in the
wave’s speed from the original speed. Give your answer to the
nearest mile per hour.
68. The cube root of a number n is the number m such
a
0
1
8
27
64
that m3 n. For example, because 23 8, the cube
3
2. The table
root of 8 is 2. You write this as 8
shows some whole numbers and their cube roots.
a. Use the table to approximate each cube root to
3
3
3
the nearest integer: 3
, 55
, 22
.
3
a
0
1
2
3
4
b. Critical Thinking Do negative numbers have
cube roots? Explain.
c. Solve the equation x 3 125.
69. Challenge Solve (x 2)2 1 37. Describe the steps you use.
Mixed Review
Write the prime factorization of the number. (Lesson 4.1)
70. 45
71. 98
72. 484
73. 700
Write the fraction in simplest form. (Lesson 4.3)
21
74. 48
13
75. 52
30
76. 125
30
77. 162
Use the percent equation to answer the question. (Lesson 7.4)
Standardized Test
Practice
78. What percent of 240 is 42?
79. What number is 80% of 60?
80. 7 is 3.5% of what number?
81. What percent of 20 is 1.3?
82. Multiple Choice What is 500
to the nearest integer?
A. 20
B. 21
C. 22
D. 23
83. Multiple Choice What is the value of the expression mn 2 when
m 4 and n 5?
F. 10
G. 20
H. 80
Lesson 9.1
I. 100
Square Roots
457