11-4
Operations With
Radical Expressions
11-4
1. Plan
Lesson Preview
Lesson Preview
✓Check Skills You’ll Need
Operations With Radical
Expressions
Lesson 11-1: Examples 1, 2, 7
Exercises 1–12, 44–51
Extra Practice, p. 712
What You’ll Learn
Check Skills You’ll Need
(For help, go to Lesson 11-1.)
OBJECTIVE
To simplify sums and
differences
1
Simplify each radical expression.
1. !52 2 "13
OBJECTIVE
To simplify products
and quotients
2
2. !200 10 "2
3. 4!54 12 "6
Rationalize each denominator.
5. !3 "1133
!11
. . . And Why
To find the width of a
painting, as in Example 6
6. !5 "410
30x
7. !15 "2x
!8
!2x
New Vocabulary • like radicals • unlike radicals • conjugates
Lesson Resources
Interactive lesson includes instant
self-check, tutorials, and activities.
OBJECTIVE
Teaching Resources
Practice, Reteaching, Enrichment
1
Part
1 1 Simplifying Sums and Differences
For radical expressions, like radicals have the same radicand. Unlike radicals do
not have the same radicand. For example, 4 !7 and 212 !7 are like radicals, but
3 !11 and 2 !5 are unlike radicals. To simplify sums and differences, you use the
Distributive Property to combine like radicals.
Reaching All Students
Practice Workbook 11-4
Spanish Practice Workbook 11-4
Basic Algebra Planning Guide 11-4
1
Presentation Assistant Plus!
Transparencies
• Check Skills You’ll Need 11-4
• Additional Examples 11-4
• Student Edition Answers 11-4
• Lesson Quiz 11-4
PH Presentation Pro CD 11-4
4. "125x2
5x "5
EXAMPLE
Combining Like Radicals
Simplify !2 + 3 !2.
!2 + 3 !2 = 1 !2 + 3 !2
= (1 + 3) !2
= 4 !2
Check Understanding
Both terms contain !2.
Use the Distributive Property to combine like radicals.
Simplify.
1 Simplify each expression.
a. 23 !5 - 4 !5 –7 "5
Computer Test Generator CD
b. !10 - 5 !10 –4 "10
You may need to simplify a radical expression to determine if you have
like radicals.
Technology
Resource Pro® CD-ROM
Computer Test Generator CD
Prentice Hall Presentation Pro CD
2
Student Site
• Teacher Web Code: aek-5500
• Self-grading Lesson Quiz
Teacher Center
• Lesson Planner
• Resources
Simplifying to Combine Like Radicals
Simplify 7 !3 - !12.
Need Help?
www.PHSchool.com
EXAMPLE
7 !3 - !12 = 7 !3 - !4 ? 3
= 7 !3 - !4 ? !3
= 7 !3 - 2 !3
Multiplication Property
of Square Roots:
!ab = !a ? !b and
!a ? !b = !ab
= (7 2 2) !3
= 5 !3
Check Understanding
Plus
600
2 Simplify each expression.
a. 3 !20 + 2 !5 8 "5
4 is a perfect square and a factor of 12.
Use the Multiplication Property of Square Roots.
Simplify !4.
Use the Distributive Property to combine like radicals.
Simplify.
b. 3 !3 - 2 !27 –3 "3
Chapter 11 Radical Expressions and Equations
✓ Ongoing Assessment and Intervention
600
Before the Lesson
During the Lesson
After the Lesson
Diagnose prerequisite skills using:
• Check Skills You’ll Need
Monitor progress using:
• Check Understanding
• Additional Examples
• Standardized Test Prep
Assess knowledge using:
• Lesson Quiz
• Computer Test Generator CD
OBJECTIVE
2. Teach
2
1 2 Simplifying Products and Quotients
Part
When simplifying a radical expression like !3 A !6 1 7B , use the Distributive
Property to multiply !3 times A !6 1 7B .
3
Math Background
Unlike radicals, such as !2 and
!3, are analogous to different
variables, such as x and y. They
cannot be combined by adding
or subtracting.
Using the Distributive Property
EXAMPLE
Simplify !3 A !6 1 7B .
!3 A !6 1 7B = !18 + 7!3
Use the Distributive Property.
= !9 ? !2 + 7!3 Use the Multiplication Property of Square Roots.
3 Simplify each radical expression.
a. !5 A 2 1 !10 B
b. !2x A !6x 2 11 B
2 "5 ± 5 "2
1
c. !5a A !5a 1 3 B
5a ± 3 "5a
2x "3 – 11"2x
If both radical expressions have two terms, you can multiply the same way you find
the product of two binomials, by using FOIL.
4
1
Simplify.
= 3!2 + 7!3
Check Understanding
OBJECTIVE
EXAMPLE
Simplifying Using FOIL
EXAMPLE
A !5 - 2 !15 B A !5 + !15 B
= !25 + !75 - 2!75 - 2!225
Additional Examples
1 Simplify 4 !3 + !3. 5 !3
Use FOIL.
= 5 - !75 - 2(15)
Combine like radicals and simplify !25 and !225.
= 5 - !25 ? !3 - 30
Use the Multiplication Property of Square Roots.
= 5 - !25 ? 3 - 30
2 Simplify 8 !5 - !45. 5 !5
25 is a perfect square factor of 75.
= 5 - 5!3 - 30
Simplify !25.
= -25 - 5!3
Simplify.
4 Simplify each radical expression.
a. A 2 !6 + 3 !3 B A !6 - 5 !3 B
–33 – 21"2
OBJECTIVE
2
4
b. A !7 +
4B2
23 ± 8 "7
Conjugates are the sum and the difference of the same two terms. The radical
expressions !5 + !2 and !5 - !2 are conjugates. The product of two
conjugates results in a difference of two squares.
Need Help?
Remember that the
difference of two
squares can be
factored as
(a + b)(a - b).
Technology Tip
Suggest that students use a
calculator to verify that the
combining of like terms is correct.
Find the value of the original
equation and compare it to the
value of the answer.
Simplify A !5 - 2!15 B A !5 + !15 B .
Check Understanding
Teaching Notes
A !5 1 !2 B A !5 2 !2 B = A !5 B 2 - A !2B 2
=5-2
=3
Teaching Notes
EXAMPLE
Inclusion
Students with some kinds of vision
problems will have difficulty
distinguishing between the terms,
especially when using FOIL, due
to the distraction of so many
radical signs. Suggest that they
add extra space between the
expressions and around each
operation sign, and use arcs to
connect each pair of terms to be
multiplied.
Notice that the product of these conjugates has no radical.
You recall that a simplified radical expression has no radical in the denominator.
When a denominator contains a sum or a difference including radical expressions,
you can rationalize the denominator by multiplying the numerator and the
denominator by the conjugate of the denominator. For example, to simplify a
6
radical expression like
, you multiply by "5 1 "2.
"5 2 "2
"5 1 "2
Additional Examples
3 Simplify !5 (!8 + 9).
2 !10 ± 9 !5
4 Simplify
( !6 - 3 !21 )(!6 + !21 ).
–57 – 6!14
Lesson 11-4 Operations With Radical Expressions
601
Reaching All Students
Below Level To emphasize to students
that only like radicals can be combined
show them !9 + !16 2 !25. Rather
!9 + !16 = 3 + 4 = 7.
Advanced Learners Have students
simplify
24
.
"x 1 "5y
English Learners
See note on page 602.
Inclusion
See note on page 601.
601
Additional Examples
5 Simplify
8
Á7 2 Á3
. 2 !7 ± 2 !3
5
Simplify
6 The ratio length : width of a
5
English Learners
The letters j and g are
pronounced differently in
different languages. Help
students pronounce the word
conjugate. Explain that conjugate
means to join together in pairs.
6
"7
"10
EXAMPLE
"5
"2
5
8 "11 2 8 "3
in.; 32 in.
EXAMPLE
Rationalizing a Denominator Using Conjugates
6
.
!5 2 !2
!5 1 !2
6
6
!5 2 !2 = !5 2 !2 ? !5 1 !2
painting is approximately equal to
the golden ratio (1 + !5 ) 2.
The length of the painting is
51 in. Find the exact width of the
painting in simplest radical form.
Then find the approximate width
to the nearest inch.
251 ( 12Á 5 )
2
EXAMPLE
Check Understanding
Multiply the numerator and the denominator
by the conjugate of the denominator.
=
6 A !5 1 !2 B
5 22
Multiply in the denominator.
=
6 A !5 1 !2 B
3
Simplify the denominator.
= 2A !5 1 !2B
Divide 6 and 3 by the common factor 3.
= 2!5 1 2!2
Simplify the expression.
5 Simplify each expression. See left.
4
24
a.
b.
!7 1 !5
!10 1 !8
c.
25
!11 2 !3
You can solve a ratio involving radical expressions.
6
Connection to Art
EXAMPLE
Real-World
Problem Solving
Art The ratio length : width of this painting by Mondrian is approximately equal
to the golden ratio A 1 + !5 B i 2. The length of the painting is 81 inches. Find the
width of the painting in simplest radical form. Then find the approximate width to
the nearest inch.
Encourage students to research
how the golden ratio was used in
Renaissance architecture. Invite
interested students to make a
presentation for the class
describing the golden ratio and
showing pictures of buildings
incorporating the golden ratio
proportions.
Define 81 = length of painting
x = width of painting
Relate A1 1 !5 B i 2 = length i width
Write
1 1 !5
= 81
2
x
xA1 1 !5 B = 162
Closure
Cross multiply.
x A 1 1 !5 B
162
=
A 1 1 !5 B
A 1 1 !5 B
A 1 2 !5 B
x = A 162 B ? A
1 1 !5
1 2 !5 B
Ask students to summarize how
to add and subtract radicals. You
remove perfect squares that are
factors of the radicands, and
write their square roots outside
the radical signs and combine
like terms.
x=
x=
162 A 1 2 !5 B
125
Divide both sides by A1 1 !5 B.
Multiply the numerator and the
denominator by the conjugate of the
denominator.
Multiply in the denominator.
162 A 1 2 !5 B
24
Simplify the denominator.
x=
281 A 1 2 !5 B
2
Divide 162 and –4 by the common factor –2.
x=
50. 06 07 5309
Use a calculator.
x < 50
The exact width of the painting is
painting is 50 inches.
Check Understanding
602
281A1 2 !5 B
2
6 Another painting has a length i width ratio approximately equal to the golden
ratio A 1 + !5 B i 2. Find the length of a painting if the width is 34 inches. 55 in.
Chapter 11 Radical Expressions and Equations
pages 603–606 Exercises
22. –9 – 14 "6
23. 58 – 10 "30
24. 11 – 4 "7
602
inches. The approximate width of the
25. 43 ± 4 "30
26. 32 ± 9 "11
27. 23 – 5 "13
EXERCISES
For more practice, see Extra Practice.
3. Practice
Practice
Practiceand
andProblem
ProblemSolving
Solving
Practice by Example
Example 1
(page 600)
Example 2
(page 600)
Assignment Guide
Simplify each expression.
1. 23!6 + 8!6 5 "6
2. 16 !10 + 2!10 18 "10 3. !5 - 3!5 –2 "5
4. 6 !7 - 4!7 2 "7
5. 15 !2 - !2 14 "2
7. !2, !32 yes
8. !3, !75 yes
10. !18 + !2 4 "2
16. !2 A !8 - 4 B 4 – 4 "2
18. 2 !3 A !3 - 1 B 6 – 2 "3
(page 602)
Example 6
(page 602)
8
28.
2"7 ± 2"3
!7 2 !3
48
30.
–4"6 – 12"2
!6 2 !18
240
32.
–5"11 – 5"3
!11 2 !3
17. !3 A !27 + 1 B 9 ± "3
19. !3 A !15 + 2 B 3 "5 ± 2 "3
Error Prevention
21. !6 A !6 - 5 B 6 – 5 "6
Exercises 10–15 Students may
think a radical cannot be
simplified because they choose
factors that do not contain a
perfect square. For example, for
!18 students might choose 6
and 3 instead of 9 and 2. Suggest
to them that they factor the
radicand completely and look
for pairs of factors.
27. A 4 - !13 B A 9 + !13 B
212
–6 "2
!8 2 !2
3"10 1 3"5
3
31.
5
!10 2 !5
9
33.
18"3 ± 9"11
!12 2 !11
29.
Alternative Method
3
35.
= 1 2x !5
1 1 !5
Exercises 20–27 Help students see
36. !2 2 1 = x2
!2 1 1
37. The ratio of the length to the width of a painting is A1 1 !5 B i 2. The length is
12 ft. What is the width? 7.4 ft
B
Apply Your Skills
Simplify each expression. 39–46. See margin.
38. !40 + !90 5 "10
41. A !3 1 !5 B 2
44. 2 !2 A 22!32 + !8 B
39. 3!2 A 2 + !6 B
42. !13 1 !10
!13 2 !5
45. 4!50 - 7!18
40. !12 + 4!75 - !36
43. A !7 + !8 B A !7 + !8 B
46. 2 !12 1 3 !6
!9 2 !6
formula
that when multiplying the square
roots of identical radicands, the
product is just the radicand. This
method is quicker than
multiplying the radicands, and
then finding the square root of
the product.
Enrichment 11-4
Reteaching 11-4
Practice 11-4
Name
Class
Practice 11-4
1. 3"7 + 5"7
2. 10"4 - "4
7. "28 + "63
8. 3"6 - 8"6
4. "45 + 2"5
10. "18 - "50
13. 3 Q 8"3 2 7 R
!m1
=
, where m1 and m2 are the masses of the molecules of the
!m2
16. "6 Q 7 1 3"3 R
5. 12"11 + 7"11
11. 4"2 + 2"8
14. 8 Q 2"5 1 5"2 R
17. 8 Q 4 2 3"2 R
26. 8"13 - 12"13
27. 13"40 + 6"10
34. "19 + 4"19
37.
1
"2 2 "3
2
43. 3 2 "6
5 2 2"6
23. 10"13 - 7"13
29. 12"29 - 15"29
32. 3"6 Q 2"3 1 "6 R
35. 12"9 - 4"9
38.
5
"7 2 "3
41. Q 3"5 1 "5 R
44.
212
"6 2 3
36. 6 – 4 "2 ; 0.3
42. 13 1 "65 1 8"130 1 5"2
43. 15 ± 4 "14
35. –43 ; –1.3
39. 6 "2 ± 6 "3
41. 8 ± 2 "15
45. –"2
46. 4 "3 ± 4 "2 ± 3 "6 ± 6
24. 12"6 - 4"24
30. 10"6 - 2"6
33. 17"35 + 2"35
36. "8 Q "2 2 7 R
3
"5 1 5
7
"2 2 "7
45. 2"3 2 "6
5"3 1 2"6
39.
2
Solve each exercise by using the golden ratio Q 1 1 " 5 R : 2.
40. 22 "3 – 6
18. 2"12 + 6"27
25. 5"7 + "28
40. Q "6 2 3 R
34. 10( "2 ± 1); 24.1
9. "3 Q "6 2 "12 R
15. 17"21 - 12"21
21. "10 Q 3 2 2"6 R
31. 8"3 - "75
44. –24
6. "2 Q 2"3 2 4"2 R
12. 13"15 2 11"15
20. 8"26 + 10"26
28. 23"3 Q "6 1 "3 R
603
3. 4"2 Q 2 1 2"3 R
19. 19"3 + "12
22. 9"2 - "50
r
gases. Find r12 if m1 = 12 units and m2 = 30 units. "510
Lesson 11-4 Operations With Radical Expressions
Date
Operations with Radical Expressions
Simplify each expression.
47. Chemistry The ratio of the rates of diffusion of two gases is given by the
r1
r2
Objective
A B Core 16–47, 55–61
C Extension 64–71
Mixed Review 76–92
Find an exact solution for each equation. Find the approximate solution to the
nearest tenth. 34–36. See margin.
34. 5 !2 = x
!2 2 1
!2
2
Standardized Test Prep 72–75
22. A 3 !2 + !3 B A !2 - 5!3 B
23. A 2!5 - !6 B A 4!5 - 3 !6 B
22–27. See margin p. 602.
2
25. A 2!10 + !3 B 2
24. A !7 - 2 B
26. A 2!11 + 5 B A !11 + 2 B
Example 5
!5, !50 no
15. 24!10 + 6!40 8 "10
20. !2 A 3 + 3!2 B 3 "2 ± 6
(page 601)
9.
13. 4!5 - 2!45 –2 "5
14. 3 !7 - !28 "7
Example 4
62–63
11. 2!12 - 7!3 –3 "3
12. !8 + 2!2 4 "2
(page 601)
Objective
A B Core 1–15, 48–54,
Tell whether each pair of expressions can be simplified to like radicals.
Simplify each expression.
Example 3
1
6. 25!3 - 3!3 –8 "3
42.
© Pearson Education, Inc. All rights reserved.
A
46. The ratio of the height ; width of a window is equal to the golden ratio.
The width of the door is 36 in. Find the height of the door. Express your
answer in simplest radical form and in inches.
47. The ratio of the length ; width of a flower garden is equal to the golden
ratio. The width of the garden is 14 ft. Find the length of the garden.
Express your answer is simplest radical form and in feet.
48. The ratio of the width ; height of the front side of a building is equal to
the golden ratio. The height of the building is 40 ft. Find the width of
the building. Express your answer in simplest radical form and in feet.
4
Lesson 11-4 Practice
Algebra 1 Chapter 11
603
Connection to History
Exercise 54 The greatest number
of kites flown on a single line is
11,284. Sadao Harada and a team
of assistants achieved this feat in
Kagoshima, Japan, in October
1990.
Math Tip
Geometry Find the exact perimeter of each figure below. 48–51. See left.
48. 8 "2 units
48.
4
49. (10 ± 10 "2 ) units
4
x
x
51. (4x ± x "10) units
⫺4 ⫺2
Exercises 55–56 Remind students
O
2
⫺4 ⫺2
4
O
⫺2
⫺2
⫺4
⫺4
x
50.
y
2
2
50. 6 "10 units
that for some kinds of problems
percents must be rounded up,
disregarding the usual rounding
rules. Show them that if they
round down they will not reach
the target amount.
49.
y
4
2
51.
x兹10
x
3兹5
x
x
x
52. Open-Ended Make up three sums that are less than or equal to 50. Use
the square roots of 2, 3, 5, or 7, and the whole numbers less than 10.
For example, 8 "5 1 9"7 # 50 . See margin.
53a. The student
simplified
"48 as 2 "24
instead
of 2 "12 or 4 "3 .
b. 2 "6 ± 4 "3
54a. 2 "2 or 2.8 ft
53. Error Analysis When simplifying "24 1 "48, a student wrote 3"24 5 6"6.
a. What error did the student make? See left.
b. Simplify "24 1 "48 correctly.
s
54. You can make a box kite like the one at the
right in the shape of a rectangular solid. The
opening at each end of the kite is a square.
a. Suppose the sides of the square are
2 ft long. How long are the diagonal
struts used for bracing? See left.
b. Suppose each side of the square has
length s. Find the length of the diagonal
struts in terms of s. Write your answer
in simplest form. s "2
s
Investments For Exercises 55–57, the formula r ≠ Î A
P – 1 gives the interest
rate r that will allow principal P to grow into amount A in two years, if the interest
is compounded annually. Use the formula to find the interest rate you would need
to meet each goal.
55. Suppose you have $500 to deposit into an account. Your goal is to have $595 in
that account at the end of the second year. 9.1%
56. Suppose you have $550 to deposit into an account. Your goal is to have $700 in
that account at the end of two years. 12.8%
57. Suppose you have $600 to deposit into an account. Your goal is to have $800 in
that account at the end of two years. 15.5%
58. a. Suppose n is an even number. Simplify "xn. x 2
n21
b. Suppose n is an odd number greater than 1. Simplify "xn. x 2 "x
n
59. Critical Thinking Simplify a"b. "bab
b"a
604
Chapter 11 Radical Expressions and Equations
pages 603–606 Exercises
604
52. Answers may vary.
Sample: 8 "2 ± 4 "3 ,
2 "7 ± 9 "3 , 6 "5 ±
3 "7
4. Assess
60. Find the value of the numerical expression for Professor Hinkle’s age
in the cartoon. about 251 years
Lesson Quiz 11-4
Simplify each expression.
1. 12 !16 - 2 !16 40
2. !20 - 4 !5 –2 !5
3. !2 (!2 + 3 !3 ) 2 ± 3 !6
4. (!3 - 2 !21 )(!3 + 3 !21 )
–123 ± 3 !7
They are unlike radicands.
61. Writing Explain why !3 + !6 cannot be simplified.
兹 the
兹 table.
兹
62. a. Copy and complete
a
b. No; the only values
it worked for were
0 and 1.
C
Challenge
b
兹a
兹b
兹a 兹b
1
0
■ 1
■0
■ 1
1
25
9
■ 1
■3
■6
■ 5
■ 8
■ 14
■9
■ 19
64
36
■ 4
■ 5
■ 8
100
81
■10
16
Á5 2 Á7
–8 !5 – 8!7
兹
16
Alternative Assessment
兹a b
■1
Divide the class into small groups.
Tell students they will be team
teachers who will teach a class
what they should learn in this
lesson. Have the group design a
problem for each of Examples 2–5
and present to the class the
problems and how to solve them.
Allow groups to use the board.
■"17
■ "34
■10
■ "181
b. Does !a + !b always equal !a 1 b? Explain. See left.
63. Error Analysis Explain the error in the work below.
!41 = !16 1 25 = !16 + !25 = 4 + 5 = 9 "a 1 b u "a ± "b
Simplify each expression.
2
64. !18 + 3 9"
!2 2
3 23"7
65. !28
21
3 +
!7
67. !27 1 !48 2 !75 2
70. 2 "2 – "6 – "3 ± 3
兹
5.
66.
3
Å5
+
5 8"15
Å3
15
68. !288 + !50 - !98 10 "2
!3
69. A !2 + !32 B A !2 + !8 + !32 B 70 70. !5 1 !10 2 "15
!10 2 !5
71. Find the length of each hypotenuse. Write your answers in simplified radical form.
a.
b.
兹10 ⫺ 兹2
兹20 ⫺ 兹6
兹20 ⫹ 兹6
2 "6
2 "13
c. If the length of the legs of a right triangle are !p + !q and !p - !q,
write an expression for the length of the hypotenuse. "2(p 1 q)
兹10 ⫹ 兹2
Standardized
Standardized
Test Prep Test Prep
Multiple Choice
72. Simplify 4!75 + !27. B
A. 12!3
B. 23!3
C. 4!102
D. 5!102
Lesson 11-4 Operations With Radical Expressions
605
605
Standardized Test Prep
Resources
Take It to the NET
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 633
• Test-Taking Strategies, p. 628
• Test-Taking Strategies with
Transparencies
Online lesson quiz at
www.PHSchool.com
Web Code: aea-1104
Short Response
Extended Response
73. Which radical expression is NOT equal to 5"2? I
F. !8 + !18
G. !98 - !8
H. 2!32 + !162
I. !48 + !2
74. Simplify A3!5 - !2 B A !5 + 5!2 B . Show your work. See back of book.
75. Explain the steps needed to simplify
5
. See back of book.
!7 1 !21
Exercise 74 Remind students to
use FOIL. Encourage them to
draw curved arrows from each
term in the first binomial to each
term in the second binomial to
help in multiplying the terms
together.
Mixed Review
Review
Mixed
Lesson 11-3
Find the distance between the points in each pair. If necessary, round to the
nearest tenth.
76. (2, 6), (8, 13) 9.2 units
77. (-1, 7), (5, 10) 6.7 units 78. (-6, 2), (20, -1)
26.2 units
Find the midpoint of each segment with the given endpoints.
79. A(4, -1) and B(2, 11) (3, 5)
Lesson 10-5
80. H(-5, 6) and K(1, 7) (–2, 6.5)
Solve each equation by factoring.
81. 5t 2 - 35t = 0 0, 7
–9, –3
–2, 9
82. p 2 - 7p - 18 = 0
83. k 2 + 12k + 27 = 0
84. y 2 - 2y = 24 –4, 6
Lesson 9-4
85. m 2 + 30 = -17m
–15, –2
Find each product. 87–89. See margin.
86. 2a 2 = -7a - 3
–3, –12
87. (b + 11)(b + 11)
88. (2p + 7)(2p + 7)
89. (5g - 7)(5g + 7)
90. (3x + 1)(3x - 1)
91. Q 13k 2 9 R Q 13k 1 9 R
92. (d - 1.1)(d - 1.1)
9x 2
–1
1 2
9k
– 81
d 2 – 2.2d ± 1.21
Algebra at Work
Auto Mechanic
Auto mechanics work to see that car engines get the most out of every
gallon of gasoline. Formulas used by mechanics often involve radicals.
For example, a car gets its power when gas and air in each cylinder are
compressed and ignited by a spark plug. An engine’s efficiency e is given
by the formula e = c 2c "c, where c is the compression ratio.
Because of the complexity of such formulas and of modern highperformance engines, today’s auto mechanic must be a highly trained and
educated professional who understands algebra, graph reading, and the
operation of computerized equipment.
Take It to the NET For more information about a career
as an auto mechanic, go to www.PHSchool.com.
Web Code: aeb-2031
606
Chapter 11 Radical Expressions and Equations
pages 603–606 Exercises
87. b2 ± 22b ± 121
88. 4p2 ± 28p ± 49
89. 25g2 – 49
606