Download section 7.3 Simplifying Radical Expressions

Chapter 7
488
Radicals and Complex Numbers
section 7.3
Objectives
1. Multiplication and Division
Properties of Radicals
2. Simplifying Radicals by
Using the Multiplication
Property of Radicals
3. Simplifying Radicals by
Using the Division Property
of Radicals
Simplifying Radical Expressions
1. Multiplication and Division Properties of Radicals
You may have already noticed certain properties of radicals involving a product or
quotient.
Multiplication and Division Properties of Radicals
n
n
Let a and b represent real numbers such that 1
a and 1
b are both real. Then
n
n
n
1. 1
ab 1
a 1
b
Multiplication property of radicals
n
2.
Concept Connections
Multiply or divide the radicals.
1. 15 16
2.
a
1a
n
n
Ab
1b
b0
Division property of radicals
Properties 1 and 2 follow from the properties of rational exponents.
1ab 1ab2 1n a1nb1n 1a 1b
n
n
n
a
a 1n
a1n
1a
a b 1n n
Ab
b
b
1b
n
n
110
15
The multiplication and division properties of radicals indicate that a product or quotient within a radicand can be written as a product or quotient of radicals, provided
the roots are real numbers. For example:
1144 116 19
25
125
A 36
136
The reverse process is also true. A product or quotient of radicals can be written
as a single radical provided the roots are real numbers and they have the same indices.
13 112 136
3
18
3
2125
8
3
A 125
In algebra it is customary to simplify radical expressions as much as possible.
Simplified Form of a Radical
Consider any radical expression where the radicand is written as a product of
prime factors. The expression is in simplified form if all the following conditions are met:
1. The radicand has no factor raised to a power greater than or equal to
the index.
2. The radicand does not contain a fraction.
3. There are no radicals in the denominator of a fraction.
Answers
1. 130
2. 12
Section 7.3
Simplifying Radical Expressions
489
2. Simplifying Radicals by Using the Multiplication
Property of Radicals
The expression 2x2 is not simplified because it fails condition 1. Because x2 is a
perfect square, 2x2 is easily simplified:
2x2 x
for x 0
However, how is an expression such as 2x9 simplified? This and many other radical expressions are simplified by using the multiplication property of radicals. The
following examples illustrate how nth powers can be removed from the radicands
of nth roots.
example 1
Using the Multiplication Property to Simplify a
Radical Expression
Skill Practice
Use the multiplication property of radicals to simplify the expression 2x .
Assume x 0.
9
Simplify the expressions.
Assume all variables represent
positive real numbers.
3. 2a3
4. 2y 31
Solution:
The expression 2x9 is equivalent to 2x8 x. Applying the multiplication property of radicals, we have
2x9 2x8 x
2x8 2x
Apply the multiplication property of radicals.
Note that x8 is a perfect square because
x8 1x4 2 2.
x4 1x
Simplify.
In Example 1, the expression x9 is not a perfect square. Therefore, to simplify
2x , it was necessary to write the expression as the product of the largest perfect
square and a remaining or “left-over” factor: 2x9 2x8 x. This process also applies to simplifying nth roots, as shown in Example 2.
9
example 2
Using the Multiplication Property to Simplify a
Radical Expression
Use the multiplication property of radicals to simplify each expression. Assume
all variables represent positive real numbers.
4
3
b. 2
w7z9
a. 2b7
Skill Practice
Simplify the expressions.
Assume all variables
represent positive real
numbers.
4 25
5. 2
v
3 8 12
6. 2
pq
Solution:
The goal is to rewrite each radicand as the product of the largest perfect square
(perfect cube, perfect fourth power, and so on) and a left-over factor.
4
4
a. 2b7 2b4 b3
4
b4 is the largest perfect fourth power in the
radicand.
4
2 b4 2 b3
Apply the multiplication property of radicals.
Answers
4
3
b2b
Simplify.
3. a1a
4
5. v 6 1v
4. y15 1y
3
6. p 2q 4 2p 2
490
Chapter 7
Radicals and Complex Numbers
3
3
b. 2w7z9 2 1w6z9 2 1w2
3
3
2
w6z9 2
w
2 3 3
w z 1w
w6z9 is the largest perfect cube in the radicand.
Apply the multiplication property of radicals.
Simplify.
Each expression in Example 2 involves a radicand that is a product of variable
factors. If a numerical factor is present, sometimes it is necessary to factor the coefficient
before simplifying the radical.
Skill Practice
Simplify the radicals. Assume
all variables represent positive
real numbers.
7. 224
8. 5218
4
9. 232a10b19
example 3
Using the Multiplication Property to Simplify Radicals
Use the multiplication property of radicals to simplify the expressions. Assume
all variables represent positive real numbers.
a. 156
b. 6250
Solution:
a. 256 223 7
Factor the radicand.
2122 2 12 72
Avoiding Mistakes: The
multiplication property of
radicals allows us to simplify a
product of factors within a
radical. For example:
2x 2y 2 2x 2 2y 2 xy
However, this rule does not
apply to terms that are added
or subtracted within the
radical. For example:
2x 2 y 2 and 2x 2 y 2
cannot be simplified.
3
c. 240x3y5z7
22 is the largest perfect square in the
radicand.
222 22 7
Apply the multiplication property of
radicals.
2114
Simplify.
Calculator Connections
A calculator can be used to support the solution to
Example 3(a). The decimal approximation for 156
and 2114 agree for the first 10 digits. This in itself
does not make 156 2114. It is the multiplication
of property of radicals that guarantees that the expressions are equal.
b. 6250 6 22 52
Factor the radicand.
6 25 22
Apply the multiplication property
of radicals.
6 5 22
Simplify.
3022
Simplify.
2
3
c. 240x3y5z7
3
22 5x y z
3
3 5 7
2 123x3y3z6 2 15y2z2
3
3 3 3 3 6
3
2
2xyz 2
5y2z
Answers
7. 2 16
8. 15 12
4
9. 2a2b4 22a2b3
2 56
2 28
2 14
7
3
2xyz2 2
5y2z
Factor the radicand.
23x3y3z6 is the largest perfect cube.
Apply the multiplication property
of radicals.
Simplify.
2 50
5 25
5
2 40
2 20
2 10
5
Section 7.3
Simplifying Radical Expressions
3. Simplifying Radicals by Using the Division
Property of Radicals
The division property of radicals indicates that a radical of a quotient can be
written as the quotient of the radicals and vice versa, provided all roots are real
numbers.
example 4
Using the Division Property to Simplify Radicals
Use the division property of radicals to simplify the expressions. Assume all variables represent positive real numbers.
a.
a7
B a3
b.
3
1
3
3
181
c.
7150
15
d.
2c5
B 32cd8
4
Solution:
a.
a7
B a3
The radicand contains a fraction. However, the
fraction can be reduced to lowest terms.
Skill Practice
Use the division property of
radicals to simplify the
expressions. Assume all
variables represent positive
real numbers.
10.
v 21
B v5
11.
12.
22300
30
13.
2a4
a2
b.
c.
Simplify the radical.
3
1
3
The expression has a radical in the denominator.
3
1
81
3
A 81
1
B27
Reduce to lowest terms.
1
3
Simplify.
3
3
7150
15
Because the radicands have a common factor, write
the expression as a single radical (division property of
radicals).
Simplify 150.
7252 2
15
52 is the largest perfect square in the radicand.
7252 22
15
Multiplication property of radicals
7 512
15
Simplify the radicals.
1
7 512
15
Reduce to lowest terms.
3
712
3
Answers
10. v 8
2 13
12.
3
11. 2
3x 5
13. 6
y
5
264
5
2
2
3
54x17y
B 2x 2y19
491
492
Chapter 7
Radicals and Complex Numbers
d.
2c5
B 32cd8
4
section 7.3
The radicand contains a fraction.
c4
B 16d8
4
Simplify the factors in the radicand.
4 4
2
c
Apply the division property of radicals.
4
2
16d8
c
2d2
Simplify.
Practice Exercises
Boost your GRADE at
mathzone.com!
• Practice Problems
• Self-Tests
• NetTutor
• e-Professors
• Videos
For the exercises in this set, assume that all variables represent positive real numbers unless otherwise stated.
Study Skills Exercise
1. When writing a radical expression, be sure that you understand the difference between an exponent on a
coefficient and an index to a radical. Write an algebraic expression for each of the following.
a. x cubed times the square root of y
b. x times the cube root of y
Review Exercises
For Exercises 2–4, simplify the expression. Write the answer with positive exponents only.
2. 1a2b4 2 1 2 a
a
b
b3
3. a
p4
q
6
b
12
1p3q2 2
4. 1x13y56 2 6
5. Write x4 7 in radical notation.
6. Write y25 in radical notation.
7. Write 2y9 by using rational exponents.
3
8. Write 2x2 by using rational exponents.
Objective 2: Simplifying Radicals by Using the Multiplication Property of Radicals
For Exercises 9–30, simplify the radicals. (See Examples 1–3.)
9. 2x11
10. 2p15
3
11. 2q7
3
12. 2r17
13. 2a5b4
14. 2c9d6
4
15. 2x8y13
4
16. 2p16q17
Section 7.3
Simplifying Radical Expressions
17. 128
18. 163
19. 180
20. 1108
3
21. 254
3
22. 2250
23. 225ab3
24. 264m5n20
25. 218a6b3
26. 272m5n2
3
27. 216x6yz3
3
28. 2192a6bc2
4
29. 280w4z7
4
30. 232p8qr5
493
Objective 3: Simplifying Radicals by Using the Division Property of Radicals
For Exercises 31–42, simplify the radicals. (See Example 4.)
31.
x3
Bx
32.
y5
By
33.
35.
50
B2
36.
98
B2
37.
39.
3
51
16
6
40.
7118
9
41.
2p7
2p3
3
1
3
3
1
24
3
51
72
12
34.
38.
42.
2q11
2q5
3
1
3
3
1
81
3
31
250
10
Mixed Exercises
For Exercises 43–58, simplify the radicals.
43. 5118
44. 2124
45. 6175
46. 818
47. 225x4y3
48. 2125p3q2
3
49. 227x2y3z4
3
50. 2108a3bc2
51.
55.
16a2b
B 2a2b4
3
250x3y
29y
4
52.
3s2t4
B 10,000
4
53.
32x
B y10
5
54.
3
16j 3
B k3
3
56.
227a4
3
28a
57. 223a14b8c31d22
58. 275u12v20w65x80
For Exercises 59–62, write a mathematical expression for the English phrase and simplify.
59. The quotient of 1 and the cube root of w6
60. The principal square root of the quotient of h and 49
61. The principal square root of the quantity k raised
to the third power
62. The cube root of 2x4
Chapter 7
494
Radicals and Complex Numbers
For Exercises 63–66, find the third side of the right triangle. Write your answer as a radical and simplify.
63.
64.
10 ft
?
6 in.
?
12 in.
8 ft
65.
?
66.
18 m
3 cm
12 m
7 cm
?
67. On a baseball diamond, the bases are 90 ft
apart. Find the exact distance from home
plate to second base. Then round to the
nearest tenth of a foot.
68. Linda is at the beach flying a kite. The kite is
directly over a sand castle 60 ft away from
Linda. If 100 ft of kite string is out (ignoring
any sag in the string), how high is the kite?
(Assume that Linda is 5 ft tall.) See figure.
2nd base
100 ft
90 ft
60 ft
5 ft
90 ft
Home plate
Expanding Your Skills
69. Tom has to travel from town A to town C across a small mountain range. He
can travel one of two routes. He can travel on a four-lane highway from A to B
and then from B to C at an average speed of 55 mph. Or he can travel on a
two-lane road directly from town A to town C, but his average speed will be
only 35 mph. If Tom is in a hurry, which route will take him to town C faster?
B
C
40 mi
70. One side of a rectangular pasture is 80 ft in length. The diagonal distance is
110 yd. If fencing costs $3.29 per foot, how much will it cost to fence the
pasture?
110 yd
80 ft
50 mi
A
Section 7.4
chapter 7
3
1. 164
1
A 16
4.
5. 254x w
7.
For Exercises 12–19, simplify the exponential expressions. Leave no negative exponents in your final answer.
12. 823
13. 1634
2. 181
4
3
495
midchapter review
In the following exercises, assume all variables represent positive real numbers.
For Exercises 1–10, simplify the expressions.
3.
Addition and Subtraction of Radicals
2
27
A 125
3
6. 220t u v
4
2 3 7
50x3y5
B x
8.
16w5z10
B 2w2
3
14. a
1 12
b
25
15. a
4 1 2
b
25
16. a
s2t4 1 2
b
r6
17. a
p2q3 1 2
b
r5
18. 1u2v3 2 16 1u3v6 2 13
19. 1x1 2y1 4 2 1x12y1 3 2
20. Explain how to simplify 12523.
2 2
3
9. 2
14r 32 3
10.
ab
B 1a b2 2
5 51
11. Explain how to simplify 2
x .
21. Rewrite the expressions, using rational exponents.
3
a. 2
7x2
b. 221a
22. Rewrite the expressions in radical notation.
a. x23
section 7.4
Addition and Subtraction of Radicals
Definition of Like Radicals
Two radical terms are said to be like radicals if they have the same index
and the same radicand.
The following are pairs of like radicals:
same index
and
3a15
same radicand
Objectives
1. Definition of Like Radicals
2. Addition and Subtraction of
Radicals
1. Definition of Like Radicals
7a15
b. 1c14 2 3
Indices and radicands are the same.