Download Multiplying and Dividing Radical Expressions

A2_3eTE07_02_374-379 10/20/05 10:46 AM Page 374
7-2
Multiplying and Dividing
Radical Expressions
7-2
1. Plan
Objectives
1
2
To multiply radical expressions
To divide radical expressions
What You’ll Learn
Check Skills You’ll Need
• To multiply radical
Find each missing factor.
expressions
Examples
• To divide radical expressions
1
2
. . . Any Why
3
4
5
6
Multiplying Radicals
Simplifying Radical
Expressions
Multiplying Radical
Expressions
Dividing Radicals
Rationalizing the
Denominator
Real-World Connection
To transform a famous
formula, as in Example 6
1
page 368
1. 150 = 5 2(j) 6
3. 48 = 4 2(j) 3
2. 54 = (j) 3(2) 3
4. x 5 = (j) 2(x) x2
5. 3a 3b 4 = (j) 3(3b) ab
6. 75a 7b 8 = (j) 2(3a) 5a3b4
New Vocabulary • rationalize the denominator
Multiplying Radical Expressions
To multiply radicals consider the following.
Math Background
!16 ? !9 = 4 ? 3 = 12 and !16 ? 9 = !144 = 12.
So !16 ? !9 = "16 ? 9.
The properties for multiplying
and dividing radicals, presented in
this lesson, both actually hold
under less restrictive conditions
n
n
on "a and "b, but more
restrictive conditions on n.
Students can investigate these
conditions using their knowledge
of imaginary numbers from
Chapter 5. Both properties aid
in simplifying radical expressions.
To simplify radical expressions
that involve fractions, write
the expressions so that no
denominator contains a radical
and no radicand contains a
fraction.
" 28 ? " 27 = -2 ? 3 = -6 and " 28 ? 27 = " 2216 = -6.
3
3
3
3
So " 28 ? " 27 = " 28 ? 27.
3
3
3
In general, the product of the principal nth roots of two numbers equals the
principal nth root of their product.
Key Concepts
Property
Multiplying Radical Expressions
If " a and " b are real numbers, then " a ? " b = " ab.
n
1
n
EXAMPLE
n
n
n
Multiplying Radicals
Multiply. Simplify if possible.
a. !2 ? !8
More Math Background: p. 366C
!2 ? !8 = !2 ? 8 = !16 = 4
b. " 25 ? " 25
3
Lesson Planning and
Resources
3
" 25 ? " 25 = " 2125 = # (25) 3 = -5
3
3
3
3
c. !22 ? !8
See p. 366E for a list of the
resources that support this lesson.
The property for multiplying radicals does not apply. !22 is not a real number.
Quick Check
PowerPoint
Bell Ringer Practice
Check Skills You’ll Need
GO for Help
374
1 Multiply. Simplify if possible.
a. "3 ? "12 6
b. " 3 ? " 29 –3
3
3
not possible
4
4
c. " 4 ? " 24
Chapter 7 Radical Functions and Rational Exponents
For intervention, direct students to:
Finding Factors
Algebra 1 Review: Page 368
Special Needs
Below Level
L1
Show students how a fraction can be rewritten in
many equivalent forms using multiplication by 1.
5
5
1
1
? A !3
B 5 !3
2 ? A 5 B 5 10
3
!3
!3
Show how a similar procedure can be applied to
radical fractions to rationalize the denominator
374
learning style: visual
L2
Have students write out the properties for multiplying
and dividing radical expressions using several
different numerical examples.
learning style: verbal
A2_3eTE07_02_374-379 10/20/05 10:46 AM Page 375
2" 3 is considered to be a simplified form of " 24. You can use the property for
multiplying radical expressions to simplify some radical expressions.
3
2
3
Guided Instruction
Simplifying Radical Expressions
EXAMPLE
Simplify each expression. Assume that all variables are positive. Then absolute
value symbols are never needed in the simplified expression.
2
a. "72x3
"72x 3 = "6 2 ? 2 ? x 2 ? x
=
b. " 80n5
"6 2x2
= 6x"2x
" a ? " b ≠ "ab
n
n
n
Simplify.
3
3
3
=
=
" 2 3n 3
3
?
" 10n 2
3
2n" 10n 2
3
Additional Examples
Factor into perfect cubes.
"a ? "b ≠ "ab
n
n
n
1 Multiply. Simplify if possible.
a. !3 ? !12 6
3
3
b. "216 ? "4 –4
c. !24 ? !16 The Property for
Multiplying Radicals does not
apply.
Simplify.
3
2 Simplify "50x 4 and "
18x 4. Assume that x is positive.
5x2"2; x"18x
3
Simplify the products of radicals as much as possible.
3
Multiplying Radical Expressions
EXAMPLE
Multiply and simplify " 54x 2y 3 ? " 5x 3y 4. Assume that all variables are positive.
3
3
n
n
"a ? "b ≠ "ab
" 54x 2y 3 ? " 5x 3y 4 = " 54x 2y 3 ? 5x 3y 4
3
3
n
3
= " 3 3x 3(y 2) 3 ? 10x 2y
3
3
3
= 3xy 2 " 10x 2y
3
2
1
3 Multiply and simplify
"25xy 8 ? "5x 4y 3 . Assume all
3
variables are positive. 5xy 3 "x2 y 2
"a ? "b ≠ "ab
n
n
2 Simplify each expression.
Assume all variables are positive.
a. "50x5 5x 2 !2 x
3
3
b. "54n 8 3n2 "2 n2
3
Factor into perfect cubes.
= " 3 3x 3(y 2) 3 ? " 10x 2y
Quick Check
Math Tip
PowerPoint
" 80n 5 = " 2 3 ? 10 ? n 3 ? n 2
Quick Check
EXAMPLE
Point out that to simplify an nth
root, you need to look for nth
powers that are factors of the
radicand. For example, if n = 2,
then look for factors of the
radicand that are perfect squares.
Factor into perfect squares.
? !2x
2. Teach
n
3
Simplify.
3 Multiply and simplify 3"7x 3 ? 2"21x 3y 2. Assume that all variables are positive.
42x3y"3
Dividing Radical Expressions
To divide radicals, consider the following.
!36
!36
6 2
6
= 65 and Å 36
25 = Ä Q 5 R = 5. So !25 =
!25
36 .
Å 25
In general, the quotient of the principal nth roots of two numbers equals the
principal nth root of their quotient.
Key Concepts
Property
Dividing Radical Expressions
a
If "a and " b are real numbers and b 2 0, then "
=
n
"b
n
n
n
na
Åb .
Lesson 7-2 Multiplying and Dividing Radical Expressions
Advanced Learners
375
English Language Learners ELL
L4
Point out to students that rationalizing a
denominator is a necessary algebraic skill in some
higher mathematics.
learning style: verbal
Review for students the different processes they are
using: multiplication, division, and simplification
(finding roots). Clarify that rationalizing the
denominator is a simplification process that involves
rewriting the radical in an equivalent form.
learning style: verbal
375
A2_3eTE07_02_374-379 10/20/05 10:46 AM Page 376
Guided Instruction
5
EXAMPLE
4
Alternative Method
EXAMPLE
Divide and simplify. Assume that all variables are positive.
3
32
a. "
3
" 24
The expression in part b is
simplified using Method II.
However, the expression can also
be simplified with Method I by
rewriting the expression as the
square root of a fraction.
3
" 32 = 3 32 = 3
" 28 = -2
3
Å 24
" 24
3
5
b. "3162x2
" 3x
PowerPoint
3
3
3
3
3
" 162x 5 = 3 162x 5 = 3
" 54x3 = " 33x3 ? 2 = " 33x3 ? " 2 = 3x" 2
3
2
" 3x 2
Ä 3x
Additional Examples
4 Divide and simplify. Assume all
variables are positive.
Quick Check
a. "3281 –3
4 Divide and simplify. Assume that all variables are positive.
"3
3
!27
"x
To rationalize the denominator of an expression, rewrite it so there are no
radicals in any denominator and no denominators in any radical.
Rationalizing the denominator of a numerical expression makes it easier to
1
calculate its decimal approximation. For example, !2
5 !2
2 and it is easier to
divide by 2 than by !2.
a. "3 "515
5
b. "x 2
"3x y
c.
x"3xy
3y
5
2
5 "10y
2y
Å 4y
3
EXAMPLE
Rationalizing the Denominator
Rationalize the denominator of each expression. Assume that all variables
are positive.
3
a. !2
!3
6 The distance d in meters that
an object will fall in t seconds is
given by d = 4.9t 2. Express t in
terms of d and rationalize the
Method 1
!2
5
!3
denominator. t ≠ "10d
7
25
Å3
2?35
6 5 !6 5 !6
?3
3
Ä 32
"32
Å3
Method 2
!2
= !2 ? !3 = !6
3
!3
!3 ? !3
Resources
• Daily Notetaking Guide 7-2 L3
• Daily Notetaking Guide 7-2—
L1
Adapted Instruction
Rewrite as a square root of
a fraction. Then make the denominator
a perfect square.
Multiply the numerator and denominator by !3
so the denominator becomes a whole number.
"x 3
b. !5xy
"x 3 = "x 3 ? !5xy = "5x 4y = x 2 !5y = x !5y
5xy
5xy
5y
!5xy
!5xy ? !5xy
Closure
Ask students to describe how to
multiply and divide two nth roots,
both of which are real numbers.
To multiply the nth roots, take the
nth root of the product of the
radicands. To divide the nth
roots, take the nth root of the
quotient of the radicands.
c.
3 2
Å 3x
3 2
Å 3x
Quick Check
376
376
4
15
4
c. " 1024x
4x3"x2
4
" 4x
3
5 Rationalize the denominator
of each expression. Assume all
variables are positive.
"5
4
b. "12x 2x"x
"3x
a. !243 3
3
8
b. "192x
4x2
3
"3x
Dividing Radicals
3
2
2 2
3
" 18x 2
= 3 2 ? 3 2x 2 = 18x
3 3 =
3x
Ä3 x
Ä 3x ? 3 x
Rewrite the fraction so the
denominator is a perfect cube.
5 Rationalize the denominator of each expression. Assume that the variables
are positive.
3
3
3 x"5y
4 "
18x2
a. Å 75 "535
b. "2x
c. "
3
5y
3x
"10xy
" 6x
Chapter 7 Radical Functions and Rational Exponents
A2_3eTE07_02_374-379 10/20/05 10:46 AM Page 377
6
Real-World
EXAMPLE
3. Practice
Connection
Einstein’s famous formula E = mc 2 relates energy E, mass m, and the speed of
light c. Express c in terms of E and m and rationalize the denominator.
E = mc 2
1 A B 1-22, 37-45, 56-59
E
c2 = m
2 A B 23-36, 46-55, 60
c =
Quick Check
Assignment Guide
E =
Åm
Em = !Em = !Em
m
"m2
Ä m2
6 The formula a = d2 relates the acceleration a of a moving object to the distance d it
t
moves in the time t. Solve the formula for t and rationalize the denominator.
t 5 "ada
Exercises
EXERCISES
Example 1
GO for
Help
(page 374)
Example 2
(page 375)
For more exercises, see Extra Skill and Word Problem Practice.
Example 3
(page 375)
2. " 4 ? " 16 4
1. !8 ? !32 16
3
3
"20x3
10.
"54y10
3
14.
" 81x2
3
"200a6b7
17. " 6 ? " 16 2 "12
3
3
3
21. 3" 5y3 ? 2" 50y4 30y2 "2y
Example 5
(page 376)
3
3
10x
28. !5 "4x
!2
!8x
4
2
31. "
4
"5
(page 377)
16. " 64x3y6
4
2y "4x3y2
3
4
3
18. "8y5 ? "40y2 8y3 "5y
20. 4!2x ? 5"6xy2 40xy "3
3
3
"56x 5y 5
" 250x 7y 3
26. 3 2
"7xy
" 2x y
3
2x2y2 "2
5x "x2y2
Rationalize the denominator of each expression. Assume that all variables
are positive. 30–34. See margin.
3
24. "48x 2 4x
y
"3xy
23. !500 10
!5
27. !x "22x
Example 6
15.
5
32. 15"60x
3"12x
35. Physics The formula F =
Gm1m2
r2
3
10. 3"3x2
3
11. 5x2"2x
GPS Guided Problem Solving
25.
3
3
x "
4x
29. "
3
"2 2
30.
33.
34.
"3xy 2
"5xy 3
Practice
Name
3
3
45x2
30. "3x
4
31. "250
5
Class
Practice 7-2
Date
L3
Multiplying and Dividing Radical Expressions
Multiply and simplify. Assume that all variables are positive.
1. "4 ? "6
4. 4"2x ? 3"8x
2. "9x2 ? "9y5
5. "xy ? "4xy
3
3
3. "
50x2z5 ? "
15y3z
6. 9"2 ? 3"y
Rationalize the denominator of each expression. Assume that all variables
are positive.
!xy
9x
x2
7.
8.
9. 3 3y
!3x
Ä 2
Å
"5x 4y
4
2x
10. "
4
"
3x2
"2x 2y 3
11.
x
Ä 8y
12.
3 3a
Ä 4b2c
Multiply. Simplify if possible. Assume that all variables are positive.
13. "4 ? "25
14. "81 ? "36
15. "3 ? "27
3
3
23 ? "
9
16. "
17. "3x ? "6x3
3
3
2xy2 ? "
4x2y7
18. "
Simplify. Assume that all variables are positive.
relates the gravitational force F between
an object of mass m1 and an object of mass m2 separated by distance r.
G is a constant known as the constant of gravitation. Solve the formula for r.
Rationalize the denominator.
"Gm1m2 F
r≠
F
12. 2a "4a2
L2
Reteaching
3 5
L3
L4
Enrichment
Å 3x
Lesson 7-2 Multiplying and Dividing Radical Expressions
9. 2x"5x
Exercise 35 You may wish to
point out that the distance r is the
distance between the centers of
the objects.
"32a5
3
12.
" 2250x6y5
Connection to Physics
22. 2" 2x2y2 ? 2" 15x5y
3
–2x2 y "30x
Divide and simplify. Assume that all variables are positive.
3
(page 376)
11.
"50x5
10a3b3"2b
–5x 2y "2y2
Multiply and simplify. Assume that all variables are positive.
3
3y3 "2y
19. "7x5 ? "42xy 9 7x3 y4 "6y
Example 4
Exercises 9–16 Remind students
that they do not need to use
absolute value signs since the
variables are positive.
4
3
3
4
4
3. " 9 ? "281 –9 4. " 8 ? " 32
6
3
3
3
3
3
3
5. !25 ? !5
6. "25 ? "
225 5 7. " 9 ? "224 –6 8. "212 ? "218
not possible
Simplify. Assume that all variables are positive. 9–12. See margin.
13.
69-75
76-95
Error Prevention!
Multiply, if possible. Then simplify.
9.
Test Prep
Mixed Review
To check students’ understanding
of key skills and concepts, go over
Exercises 25, 35, 36, 55, 57, 59.
377
19. "36x3
3
20. "
125y2z4
21. "18k6
3
216a12
22. "
23. "x2y10z
4
256s7t12
24. "
3
216x4y3
25. "
26. "75r3
4
27. "
625u5v8
Divide and simplify. Assume that all variables are positive.
© Pearson Education, Inc. All rights reserved.
Practice by Example
61-68
Homework Quick Check
Practice and Problem Solving
A
C Challenge
28. "6x
"3x
3
2
29. "34x
"x
31.
32.
"(2x) 2
"(5y) 4
3
"
18y2
3
"
12y
30.
33.
4 243k3
Å 3k7
162a
Ä 6a3
34. The volume of a sphere of radius r is V 5 34 pr3.
a. Use the formula to find r in terms of V. Rationalize the
denominator.
b. Use your answer to part (a) to find the radius of a sphere with
volume 100 cubic inches. Round to the nearest hundredth.
32. 5x2 "5
33.
"15y
5y
10
34. x"
2y
377
A2_3eTE07_02_374-379 10/20/05 10:46 AM Page 378
4. Assess & Reteach
B
Apply Your Skills
2 1 !3
36. a. Simplify !
by multiplying the numerator and denominator by !75 .
!75
b. Simplify the expression in (a) by multiplying by !3 instead of !75 .
2 1 !3
c. Explain how you would simplify !
. a–c. See margin.
PowerPoint
!98
Lesson Quiz
Simplify each expression. Rationalize all denominators. Assume that all variables
are positive.
3x6y5 "2y
Assume that all variables are
positive.
1. Multiply. Simplify if possible.
a. !5 ? !45 15
b. "4 ? "2000 20
3
GO
3
Visit: PHSchool.com
Web Code: age-0702
2. Simplify.
a. "8x5 2x2 !2 x
b. "2243x3y10 –3xy3 "9y
3
3
3. Multiply and simplify.
a. "18x3 ? "2x2y 3
6x2 y !xy
b. "10x2y4 ? "4x2y
3
nline
Homework Help
3
40.
20x2y 3 "y
41. 10 ± 7 "2
42. 15 ± 3 "21
8x"y
y
3
3
270x 3" y
b. "
3
y
"10xy2
b. "3 x
3 2
"4
" 2 x2
2
3
Real-World
Connection
A satellite being launched
from the cargo bay of the
space shuttle
C
Challenge
3
4
46. "5x2 3 x"10y
2y 2
"2x y
3
3
3 !7x
3
3
2xy2
50. 3" 142 "xy
" 7x y
3
10
2"25x
3
x
" 5x 2
42. 3(5 1 !21)
3
51.
3
3"11x 3y
22"12x 4y
3
"33x
24x
3
58. Physics A freely falling object hit the ground in "18a5 seconds. It fell h feet.
Use the formula h = 16t 2 to find h in terms of a. 288 a5 ft
59. Writing Does "x3 5 " x2 for all, some, or no values of x? Explain.
See margin.
60. Open-Ended Of the equivalent expressions Î 23, !2
, and !6
3 , which do you
!3
prefer to use for finding a decimal approximation with a calculator? Justify
your reasoning. Check students’ work.
3
Simplify each expression. Rationalize all denominators. Assume that all variables
are positive.
61. Î Í 16x4y4 2xy
3 2
3
x y
64. " x21y22 "xy
67. a ≠ –2c, b ≠ –6d
3
63. Î Í 8000 2 "5
62. Î Í 64x6y12 2xy2
3
5
5
65. " x24y "xyx
66.
67. Critical Thinking When "xayb is simplified, the result is
23
6y
Äx
24
6 4 3
"
x y
y
1
, where
x cy 3d
c and d
are positive integers. Express a in terms of c, and b in terms of d.
36a. "6151 3
378
41. !2A !50 1 7B
!22 ? !28 5 !22(28) 5 !16 5 4 56–57. See margin.
Have students work in pairs.
Each student should make up
a problem similar to those in
each of Examples 1–5. Then
have students work each other’s
problems and check each
other’s work.
"2 # 2 1 "3 # 2
≠ 2 114"6.
7"2 # 2
45. " 3x2 ? " x2 ? " 9x3
3
3x 2"x
3
2
1
"
3x
48. 3
" 9x 3x
3
57. Error Analysis Explain the error in this simplification of radical expressions.
Alternative Assessment
b. "6151 3
c. Answers may vary.
Sample: First simplify
the denominator.
Since "98 ≠
"2.49 ≠ 7"2, to
rationalize the
denominator, multiply
the fraction by "2.
"2
This yields
44. " 2x ? " 4 ? " 2x2
3
2x "2
5 !2 5"14x
47.
21x
3
56. Geometry A rectangular shelf is !440 cm by !20 cm. Find its area.
5. Rationalize the denominator
of each expression.
a. "7x "21x
3
"3
43. !5( !5 1 !15)
40. 5"2xy6 ? 2"2x3y
3
4. Divide and simplify.
3
39. "x5y5 ? 3"2x7y6
3
52. 22 Q " 32 1 " 54 R
53. 3 1 !5 3"55 1 5
54. !3 2 !2 "642 2
!5
!8
3
3
–4 "4 – 6"2
55. Satellites The circular velocity v, in miles per hour, of a satellite orbiting
GPS
12
Earth is given by the formula v 5 Î 1.24 3r 10 , where r is the distance
in miles from the satellite to the center of Earth. How much greater is the
velocity of a satellite orbiting at an altitude of 100 mi than one orbiting at an
altitude of 200 mi? (The radius of Earth is 3950 mi.) 212 mi/h greater
43. 5 ± 5 "3
3
"2xy
38. " 4 ? " 80 4 "5
49.
2xy "5 xy2
a. "128x
37. !5 ? !40 10 "2
68. Critical Thinking In Example 3 you saw that " 54x2y3 ? " 5x3y4 simplifies to
3
3
3xy2 " 10x2y, if you assume that all the variables are positive. Now assume that
the variables represent any real numbers. What changes must be made in the
answer? Explain. See margin.
3
378
Chapter 7 Radical Functions and Rational Exponents
56. 20 "22 cm3
57. A product of two square
roots can be simplified in
this way only if the square
roots are real numbers.
"22 and "28 are not.
59. For some values, it is
easy to see that the
equation is true if x ≠ 0
or x ≠ 1. But when
x R 0, "x3 is not a real
3
number, although "x2 is.
68. No changes need to be
made; since they are both
odd roots, there is no
need for absolute value
symbols.
A2_3eTE07_02_374-379 10/20/05 10:46 AM Page 379
Test Prep
Test Prep
Multiple Choice
69. Which expression does NOT simplify to -10? C
A. 2" 1000
Resources
B. !25 ? "28
3
C. 2!25 ? "232
3
D. "2125 ? "16
5
3
4
3
5 with a rationalized denominator? H
70. How can you write Å 2xy
3
5
F. "
2xy
3
20
G. "
2xy
" 20x 2y 2
2xy
3
H.
" 4x 2y 2
2xy
3
J.
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 427
• Test-Taking Strategies, p. 422
• Test-Taking Strategies with
Transparencies
71. What is the simplified form of 3 2 !5? A
!5
A. 3 !55 2 5
"15
C. 3 !3 2
5
B. 5 !35 2 5
D. 14 256 !5
3
72. To rationalize the denominator of Å 92, by what number would you multiply
the numerator and denominator of the fraction? G
F. 2
G. 3
H. 6
J. 9
73. Which of the following expressions is in simplest form? D
A. "20x 3
Short Response
Extended Response
B. " 81x
3
C.
6
Å2
D. !2
5
74. [2] " x is a real number
if x L 0 and "2x is a
real number if x K 0.
So the only value that
makes " x "2x a
real number is x ≠ 0.
[1] answer only OR error
describing value(s) of
variables
74. For what values of x is !x ? !2x a real number? Explain. 74–75. See
margin.
3 3
75. Rationalize the denominator of Å 2x
. Explain your steps.
75. [4] You should multiply
3 by "4x 2 because
3
Ä 2x
"4x 2
3
3
3 "4x 2 ≠
3
Ä 2x "
4x 2
3
3
Mixed Review
" 3 "4x 2
3
≠
3
"2x "4x 2
3
GO for
Help
Lesson 7-1
Simplify each radical expression. Use absolute value symbols as needed.
76. "121a90 11»a 45…
78. " 264a81 –4a27
3
80. "0.25x6 0.5»x3…
82. " 16x36y96 2»x9…y 24
4
Lesson 6-3
Lesson 5-7
3
2
"12x 2
≠ "12x ,
3
2
2x
"8x
77. 2"81c48d64 –9c 24d 32
3
79. " 32y25 2y 5
5
which has a
denominator without
a radical.
81. " x14y35 x 2y 5
7
83. "0.0064x40 0.08x20
[3] appropriate methods,
but with one minor
error
Divide. Tell whether each divisor is a factor of the dividend.
84. (y 3 - 64) 4 (y + 4)
85. (x 3 + 27) 4 (x + 3)
86. (6a 3 + a 2 - a + 4) 4 (2a + 1)
84–87. See margin.
Complete the square.
87. (2x4 - 3x 3 - 4x + 10) 4 (x - 2)
88. x 2 + 10x + j 25
89. x 2 - 10x + j 25
90. x 2 + 11x + j 121
4
91. x 2 - 11x + j
1
92. x 2 - x3 + j 36
93. x 2 + 0.3x + j 0.0225
9
94. x 2 - 34 x + j 64
9
95. x 2 + 35 x + j 100
lesson quiz, PHSchool.com, Web Code: aga-0702
3
[1] correct final
expression, but no
work shown
121
4
Lesson 7-2 Multiplying and Dividing Radical Expressions
[2] major error, but
subsequent steps
consistent with that
error
84. y 2 – 4y ± 16, R –128,
not a factor
85. x2 – 3x 9, a factor
379
86. 3a2 – 8a – 2, R 6, not a
factor
87. 2x3 ± x2 ± 2x, R 10, not a
factor
379