10.1
Goal
Simplify square roots.
Key Words
Simplifying Square Roots
Square roots are written with a radical symbol 兹m
苶. An expression
written with a radical symbol is called a radical expression, or radical .
The number or expression inside the radical symbol is the radicand .
• radical
radical symbol
• radicand
radicand
兹2
苶5
苶
radical
The radical symbol always indicates the nonnegative square root of
a number. For example, 兹2
苶5
苶 ⫽ 5 because 52 ⫽ 25.
Use a Calculator to Find Square Roots
1
EXAMPLE
Find the square root of 52. Round your answer to the nearest tenth.
Check that your answer is reasonable.
Solution
Calculator keystrokes
52
or
52
Display
Rounded value
兹5
苶2
苶 ≈ 7.2
7.21110
This is reasonable, because 52 is between the perfect squares
苶2
苶 should be between 兹4
苶9
苶 and 兹6
苶4
苶, or 7 and 8.
49 and 64. So, 兹5
The answer 7.2 is between 7 and 8.
EXAMPLE
Find Side Lengths
2
Use the Pythagorean Theorem to find the
length of the hypotenuse to the nearest tenth.
兹苵2
c
Student Help
STUDY TIP
Recall that for any
number a ≥ 0,
(兹a苶)2 ⫽ a.
兹苵3
Solution
(兹2苶 )
2
a2 ⫹ b2 ⫽ c2
Write Pythagorean Theorem.
2
⫹ (兹3
苶) ⫽ c
2
Substitute 兹2
苶 for a and 兹3
苶 for b.
2⫹3⫽c
2
Simplify.
5⫽c
2
Add.
兹5
苶⫽c
Take the square root of each side.
2.2 ≈ c
Use a calculator.
10.1
Simplifying Square Roots
537
Multiplying Radicals You can use the Product Property of Radicals to
multiply radical expressions.
Student Help
SKILLS REVIEW
To review the Product
Property of Radicals,
see p. 669.
兹a
苶 p 兹b
苶 ⫽ 兹a
苶b
苶, where a ≥ 0 and b ≥ 0.
EXAMPLE
3
Multiply Radicals
Multiply the radicals. Then simplify if possible.
a. 兹3
苶 p 兹7
苶
b. 兹2
苶 p 兹8
苶
Solution
a. 兹3
苶 p 兹7
苶 ⫽ 兹3
苶苶p苶
7
b. 兹2
苶 p 兹8
苶 ⫽ 兹2
苶苶p苶
8
⫽ 兹2
苶1
苶
⫽ 兹1
苶6
苶
⫽4
Simplifying Radicals You can also use the Product Property of
Radicals to simplify radical expressions.
兹a
苶b
苶 ⫽ 兹a
苶 p 兹b
苶, where a ≥ 0 and b ≥ 0.
To factor the radicand, look for perfect square factors.
EXAMPLE
4
Simplify Radicals
Simplify the radical expression.
a. 兹1
苶2
苶
Student Help
STUDY TIP
When you factor a
radicand, write the
perfect square factors
first.
b. 兹4
苶5
苶
Solution
a. 兹1
苶2
苶 ⫽ 兹4
苶苶p苶
3
b. 兹4
苶5
苶 ⫽ 兹9
苶苶p苶
5
⫽ 兹4
苶 p 兹3
苶
⫽ 兹9
苶 p 兹5
苶
⫽ 2兹3
苶
⫽ 3兹5
苶
Evaluate, Multiply, and Simplify Radicals
Find the square root. Round your answer to the nearest tenth. Check
that your answer is reasonable.
1. 兹2
苶7
苶
2. 兹4
苶6
苶
3. 兹8
苶
4. 兹9
苶7
苶
Multiply the radicals. Then simplify if possible.
5. 兹3
苶 p 兹5
苶
6. 兹1
苶1
苶 p 兹6
苶
7. 兹3
苶 p 兹2
苶7
苶
8. 5兹3
苶 p 兹3
苶
Simplify the radical expression.
9. 兹2
苶0
苶
538
Chapter 10
10. 兹8
苶
Right Triangles and Trigonometry
11. 兹7
苶5
苶
12. 兹1
苶1
苶2
苶
10.1 Exercises
Guided Practice
Vocabulary Check
1. What is the radicand in the expression 兹2
苶5
苶?
Match the radical expression with its simplified form.
Skill Check
2. 兹3
苶6
苶
A. 兹6
苶
3. 兹3
苶 p 兹2
苶
B. 3兹2
苶
4. 兹3
苶 p 兹6
苶
C. 4兹2
苶
5. 兹3
苶2
苶
D. 6
Use the figure shown at the right.
6. Use the Pythagorean Theorem to find the
length of the hypotenuse in radical form.
c
2
7. Use a calculator to find the length of the
4
hypotenuse to the nearest tenth.
Simplify the expression.
8. 兹4
苶9
苶
9. 兹2
苶8
苶
10. 兹7
苶2
苶
11. 兹5
苶4
苶
Practice and Applications
Extra Practice
See p. 693.
Finding Square Roots Find the square root. Round your answer to
the nearest tenth. Check that your answer is reasonable.
12. 兹1
苶3
苶
13. 兹6
苶
14. 兹9
苶1
苶
15. 兹3
苶4
苶
16. 兹1
苶0
苶6
苶
17. 兹1
苶4
苶8
苶
18. 兹6
苶2
苶
19. 兹1
苶8
苶6
苶
Pythagorean Theorem Find the length of the hypotenuse. Write your
answer in radical form.
20.
21.
兹苵2
22.
c
c
兹苵5
6
兹苵19苵
兹苵13苵
兹苵10苵
c
Homework Help
Example 1:
Example 2:
Example 3:
Example 4:
Exs. 12–19
Exs. 20–25
Exs. 26–34
Exs. 35–45
Pythagorean Theorem Find the missing side length of the right
triangle. Round your answer to the nearest tenth.
23.
x
24.
1
兹苵11苵
25.
兹苵13苵
6
x
4
x
兹苵5
10.1
Simplifying Square Roots
539
Multiplying Radicals Multiply the radicals. Then simplify if possible.
26. 兹7
苶 p 兹2
苶
27. 兹5
苶 p 兹5
苶
28. 兹3
苶 p 兹1
苶1
苶
29. 2兹5
苶 p 兹7
苶
30. 兹1
苶0
苶 p 4兹3
苶
31. 兹1
苶1
苶 p 兹2
苶2
苶
Square a Radical
EXAMPLE
Evaluate the expression.
a. (3兹7
苶 )2
b. (2兹1
苶1
苶 )2
Solution
a. (3兹7
苶 )2 ⫽ 3兹7
苶 p 3兹7
苶
b. (2兹1
苶1
苶 )2 ⫽ 2兹1
苶1
苶 p 2兹1
苶1
苶
⫽ 3 p 3 p 兹7
苶 p 兹7
苶
⫽ 2 p 2 p 兹1
苶1
苶 p 兹1
苶1
苶
⫽9p7
⫽ 4 p 11
⫽ 63
⫽ 44
Squaring Radicals Evaluate the expression. Use the example above
as a model.
32. (6兹5
苶 )2
33. (5兹3
苶 )2
34. (7兹2
苶 )2
Simplifying Radicals Simplify the radical expression.
35. 兹1
苶8
苶
36. 兹5
苶0
苶
37. 兹4
苶8
苶
38. 兹6
苶0
苶
39. 兹5
苶6
苶
40. 兹1
苶2
苶5
苶
41. 兹2
苶0
苶0
苶
42. 兹1
苶6
苶2
苶
43. 兹4
苶4
苶
You be the Judge
Determine whether the expression can be
simplified further. If so, explain how you would do so.
44. 兹8
苶0
苶 ⫽ 兹4
苶苶p苶0
2苶
45. 兹8
苶 p 兹1
苶2
苶 ⫽ 兹8
苶苶p苶2
1苶
⫽ 兹4
苶 p 兹2
苶0
苶
⫽ 兹4
苶苶p苶
2苶p苶
4苶p苶
3
⫽ 2兹2
苶0
苶
⫽ 4兹6
苶
Area Formula Use the area formula A ⴝ lw to find the area of the
rectangle. Round your answer to the nearest tenth.
46.
47.
兹苵10苵
兹苵14苵
HOMEWORK HELP
Extra help with problem
solving in Exs. 46–51 is
at classzone.com
9兹苵2
49.
50.
Chapter 10
51.
Right Triangles and Trigonometry
兹苵1苵4
4兹苵6
4兹苵3
4兹苵3
540
4兹苵
6
8兹苵3
Student Help
ICLASSZONE.COM
48.
2兹苵5
8兹苵3
3兹苵2
52. Area of an Equilateral Triangle The area of an
equilateral triangle with side length s is
given by the formula
30 ft
1
4
A ⫽ ᎏᎏs 2 兹3
苶.
30 ft
The flower bed shown is an equilateral triangle
with a side length of 30 feet. Find its area.
30 ft
53. Challenge An equilateral triangle has an area of 1 square
meter. What is the length of each side? Round your answer to
the nearest centimeter.
Standardized Test
Practice
54. Multiple Choice Which number is a perfect square?
A

B

44
C

110
D

169
500
55. Multiple Choice 兹2
苶2
苶0
苶 is between which two integers?
F

12 and 13
G

H

13 and 14
14 and 15
J

15 and 16
56. Multiple Choice Which of the following
expressions could not be used to represent
the length of the hypotenuse in the triangle
shown at the right?
A

Mixed Review
B

2兹2
苶6
苶
C

兹1
苶0
苶4
苶
2兹苵10苵
8
about 10.2
D

6兹1
苶0
苶
Finding Angle Measures Find the measure of a1. (Lesson 4.2)
57.
58.
59.
61ⴗ
45ⴗ
87ⴗ
1
51ⴗ
1
25ⴗ
1
Isosceles Triangles Find the value of x. (Lesson 4.3)
60.
61.
9
62.
xⴚ4
2x
64ⴗ
Algebra Skills
xⴙ3
4xⴗ
Distributive Property Use the distributive property to rewrite the
expression without parentheses. (Skills Review, p. 671)
63. x(x ⫹ 5)
64. 4(2x ⫺ 1)
65. x(3x ⫹ 4)
66. 5x(x ⫹ 2)
67. ⫺3(1 ⫺ x)
68. 2x ⫺ (x ⫺ 6)
10.1
Simplifying Square Roots
541