Download 9.3 operations with radicals

9.3
Operations with Radicals
(9–21)
487
OPERATIONS WITH RADICALS
9.3
In this section we will use the ideas of Section 9.1 in performing arithmetic operations with radical expressions.
In this
section
●
Adding and Subtracting
Radicals
●
Multiplying Radicals
●
Conjugates
Adding and Subtracting Radicals
To find the sum of 2
and 3, we can use a calculator to get 2 1.414 and
1.732. (The symbol means “is approximately equal to.”) We can then add
3
the decimal numbers and get
3
1.414 1.732 3.146.
2
We cannot write an exact decimal form for 2
3
; the number 3.146 is an ap. To represent the exact value of 2 3
, we just use
proximation of 2 3
3
. This form cannot be simplified any further. However, a sum of
the form 2
like radicals can be simplified. Like radicals are radicals that have the same index
and the same radicand.
, we can use the fact that 3x 5x 8x is
To simplify the sum 32 52
82
. So
true for any value of x. Substituting 2 for x gives us 32 52
like radicals can be combined just as like terms are combined.
E X A M P L E
1
Adding and subtracting like radicals
Simplify the following expressions. Assume the variables represent positive
numbers.
4
4
45
b) 6
w
w
a) 35
3
3
3
3
c) 3
5
43
65
d) 36x
2
x 6x
x
Solution
4
4
4
a) 35
45
75
b) 6
w
5
w
w
c) 3
5
43
65
33 75
Only like radicals are combined.
3
3
3
3
3
3
d) 36x
2x 6x
x 46x
3x
■
Remember that only radicals with the same index and same radicand can be
combined by addition or subtraction. If the radicals are not in simplified form, then
they must be simplified before you can determine whether they can be combined.
E X A M P L E
2
Simplifying radicals before combining
Perform the indicated operations. Assume the variables represent positive numbers.
1
18
b) 20
a) 8
5
c) 2x 4x 5
18x
3
2
3
d) 16x 4y3 54x 4y3
3
Solution
a) 8
18
4
2
9
2
22
32
Simplify each radical.
52
Add like radicals.
Note that 8 18
26
.
3
488
(9–22)
Chapter 9
calculator
Radicals and Rational Exponents
b)
15 20 55 25
15 15 55 55 and 20 25
Because
5 105
5
5
close-up
Use the LCD of 5.
115
5
3
2
c) 2x
4x 5
18x3 x2 2x
2x 5 9x2 2x
Check that
8 + 18
= 52.
x2x
2x 15x 2x
Simplify each radical.
16x2x
2x
Add like radicals only.
d) 16x y 54x y 8x y 2x
27x y 2x
3
4 3
4 3
3
3
3 3
3
3 3
3
3
3
2 xy
3xy
2x
2x
3
Simplify each radical.
■
xy2x
3
Multiplying Radicals
We have already multiplied radicals in Section 9.2, when we rationalized denomin
n
n
a b , allows multiplication of
ab
nators. The product rule for radicals, radicals with the same index, such as
15
,
5 3
3
3
3
2 5 ,
10
and
5
5
5
x2 x x3.
CAUTION
The product rule does not allow multiplication of radicals that
3
and 5.
have different indices. We cannot use the product rule to multiply 2
E X A M P L E
3
Multiplying radicals with the same index
Multiply and simplify the following expressions. Assume the variables represent
positive numbers.
43
b) 3a2 6a
a) 56
3
3
c) 4 4
helpful
hint
Students often write
15
15
22
5 15.
Although this is correct, you
should get used to the idea
that
4
x3
2
Solution
a) 56
43
5 4 6
3
2018
Product rule for radicals
20 32
18
9
2
32
602
b) 3a2 6a
18a3
Product rule for radicals
9a 2a
2
3a2a
15
15
15.
Because of the definition of a
square root, a
a
a for
any positive number a.
d)
Simplify.
c) 4 4 16
3
3
3
3
3
8 2
22
3
Simplify.
4
x2
4
9.3
x2 x4 x8
3
d)
2
489
5
4
4
(9–23)
Operations with Radicals
4
Product rule for radicals
4
4
x4 x
4
8
Simplify
4
x
x
4
8
4
4
x
x · 2
4
4
8 · 2
Rationalize the denominator.
4
x
2x
2
4
4
4
8 2 2
16
■
We find a product such as 32(42 3) by using the distributive property as we do when multiplying a monomial and a binomial. A product such as
(23 5)(33 25) can be found by using FOIL as we do for the product
of two binomials.
E X A M P L E
4
Multiplying radicals
Multiply and simplify.
a) 32 (42 3)
3
3
3
b) a (
a a2)
c) (23 5)(33 25)
d) (3 x 9)
2
Solution
a) 32 (42 3) 32 42 32 3
12 2 36
24 36
3
3
b) a (a a2 ) a2 a3
3
3
3
Distributive property
Because 2 2 2
and 2 3 6
Distributive property
a a
3
2
c) (23 5)(33 25)
F
O
I
L
23 33 23 25 5 33 5 25
18 415
315
10
15 Combine like radicals.
8 d) To square a sum, we use (a b)2 a 2 2ab b 2:
2
2
(3 x 9) 32 2 3
x 9 (
x 9)
9 6
x9x9
x9
x 6
In the next example we multiply radicals that have different indices.
■
490
(9–24)
Chapter 9
E X A M P L E
5
Radicals and Rational Exponents
Multiplying radicals with different indices
Write each product as a single radical expression.
3
4
a) 2 2
Solution
3
4
a) 2 2 213 214
2712
12 7
2
12
128
3
13
b) 2 3 2 312
226 336
6
6
22 33
6
22 33
6
108
calculator
close-up
Check that
2 2 12
8.
3
4
3
b) 2 3
12
Write in exponential notation.
Product rule for exponents: 1 1 7
3
4
12
Write in radical notation.
Write in exponential notation.
Write the exponents with the LCD of 6.
Write in radical notation.
Product rule for radicals
22 33 4 27 108
■
Because the bases in 213 214 are identical, we can add the
exponents [Example 5(a)]. Because the bases in 226 336 are not the same, we cannot add the exponents [Example 5(b)]. Instead, we write each factor as a sixth root
and use the product rule for radicals.
CAUTION
Conjugates
helpful
hint
The word “conjugate” is used
in many contexts in mathematics. According to the
dictionary, conjugate means
joined together, especially as
in a pair.
E X A M P L E
6
Recall the special product rule (a b)(a b) a 2 b 2. The product of the sum
4 3 and the difference 4 3 can be found by using this rule:
(4 3)(4 3) 42 (3)2 16 3 13
The product of the irrational number 4 3 and the irrational number 4 3 is
the rational number 13. For this reason the expressions 4 3 and 4 3 are
called conjugates of one another. We will use conjugates in Section 9.4 to rationalize some denominators.
Multiplying conjugates
Find the products. Assume the variables represent positive real numbers.
a) (2 35 )(2 35 )
b) (3 2
)(3
2
)
c) (2x
y )(2x
y )
Solution
2
a) (2 35 )(2 35 ) 22 (35 ) (a b)(a b) a 2 b 2
4 45
(35 )2 9 5 45
41
)(3
2
) 3 2
b) (3 2
1
c) (2x
y )(2x
y ) 2x y
■
9.3
Operations with Radicals
(9–25)
491
WARM-UPS
True or false? Explain your answer.
1. 3 3 6
2. 8 2 32
3. 23 33
63
3
3
4. 2 2 2
5. 25 32
610
6. 25
35
510
7. 2
(3 2
) 6
2
8. 12
26
2
9. (2 3
) 2 3
10. (3 2
)(3 2
) 1
9.3
EXERCISES
Reading and Writing After reading this section, write out
the answers to these questions. Use complete sentences.
1. What are like radicals?
2. How do we combine like radicals?
3. Does the product rule allow multiplication of unlike
radicals?
4. How do we multiply radicals of different indices?
All variables in the following exercises represent positive
numbers.
Simplify the sums and differences. Give exact answers. See
Example 1.
5. 3 23
6. 5 35
7. 57x
47x
8. 36a
76a
3
3
9. 2
2 3
2
3
3
10. 4 4
4
11. 3
5
33
5
12. 2 53
72
93
3
3
3
3
13. 2 x 2 4
x
3
3
3
3
14. 5y 4
5y x x
3
3
15. x 2x
x
3
3
16. ab a
5a
ab
Simplify each expression. Give exact answers. See Example 2.
17. 8 28
18. 12
24
19. 8 18
20. 12
27
21. 45
20
22. 50
32
23. 2 8
24. 20
125
2
25. 2
2
3
26. 3
3
1
27. 80
5
1
28. 32
2
29. 45x
3 18x
2 50x
2 20x
3
30. 12x
5 18x
300x
5 98x
31.
32.
33.
34.
3
3
24
81
3
3
24
375
4
4
48
243
5
5
64
2
492
(9–26)
Chapter 9
Radicals and Rational Exponents
3
3
35. 54t4y3 16t4y3
3
3
36. 2000w2
z5 16w2z5
Simplify the products. Give exact answers. See Examples 3
and 4.
37. 3
5
38. 5 7
39. 25 310
)
40. (32)(410
41. 27a
32a
55
42. 25c
4
4
43. 9 27
3
3
5 100
44. 2
45. (23 )
)2
46. (42
2
4x
2x2
47. 3 3 3
3
4x
4x
48. 5
25
2
4
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
3
4
23
(6 33
)
(3 35
)
25
(10
2)
5
6
(15
1)
3
3
3 2
3t
(9t
t)
3
3
3
2(12x
)
2x
(3 2)(3 5)
(5 2)(5 6)
(11
3)(11
3)
(2 5)(2 5)
(25 7)(25 4)
(26 3)(26 4)
(23 6)(3 26)
(33 2)(2 3)
Write each product as a single radical expression. See
Example 5.
3
4
63. 3 3
64. 3 3
3
4
3
5
65. 5 5
66. 2 2
3
3
67. 2 5
68. 6 2
3
4
3
4
69. 2 3
70. 3 2
Find the product of each pair of conjugates. See Example 6.
71. (3 2)(3 2)
72. (7 3
)(7 3
)
73. (5 2
)(5 2
)
74. (6 5
)(6 5
)
75. (25 1)(25 1)
76. (32 4)(32 4)
77. (32 5
)(32
5
)
78. (23 7
)(23 7
)
79. (5 3x )(5 3x )
80. (4y 3z )(4y 3z )
Simplify each expression.
81. 300
3
82. 50
2
83. 25 56
84. 36 510
)(7 2)
85. (3 27
86. (2 7
)(7 2)
4w
87. 4w
5m
88. 3m
3x 3 6x 2
89. 90. 2t 5 10t 4
1
1
1
91. 2 8 18
1
92. 3
13 3
93. (25 2
)(35 2
)
)(22 33
)
94. (32 3
2 2
95. 3
5
2 3
96. 4
5
97. (5 22
)(5 22
)
)(3 27
)
98. (3 27
2
99. (3 x)
2
100. (1 x)
2
101. (5x 3)
2
102. (3a 2)
2
103. (1 x
2)
2
104. (x
1 1)
105. 4w
9w
106. 10m
16m
3
107. 2
a 3
a3 2a4a
2
w y 7
w2y 6
w2y
108. 5
109. x5 2x
x3
3
110. 8x 50x 3 x
2x
3
3
4
111. 16x 5x
54x
3
3
5 7
112. 3x 5y 7 24x
y
4yx
16
114. 9
z
x
x
115. 5
5
7
113.
3
4
3
3
3
5
9.4
4
4
4
116. a3(
a a5 )
3
117. 2x 2x
3
4
118. 2m
2n
More Operations with Radicals
(9–27)
493
122. Area of a triangle. Find the exact area of a triangle
meters and a height of 6
meters.
with a base of 30
In Exercises 119–122, solve each problem.
119. Area of a rectangle. Find the exact area of a rectangle
feet.
that has a length of 6 feet and a width of 3
––
√6 m
120. Volume of a cube. Find the exact volume of a cube
with sides of length 3
meters.
—–
√30 m
FIGURE FOR EXERCISE 122
––
√3 m
GET TING MORE INVOLVED
––
√3 m
––
√3 m
FIGURE FOR EXERCISE 120
121. Area of a trapezoid. Find the exact area of a trapezoid
with a height of 6 feet and bases of 3 feet and
12
feet.
–
√ 3 ft
––
√ 6 ft
—–
√12 ft
FIGURE FOR EXERCISE 121
9.4
In this
section
●
Dividing Radicals
●
Rationalizing the
Denominator
●
Powers of Radical
Expressions
123. Discussion. Is a
b
a
b for all values
of a and b?
124. Discussion. Which of the following equations are
identities? Explain your answers.
a) 9x
3x
b) 9
x 3 x
c) x
4 x 2
x
x
d) 2
4
125. Exploration. Because 3 is the square of 3, a binomial such as y 2 3 is a difference of two squares.
a) Factor y 2 3 and 2a 2 7 using radicals.
b) Use factoring with radicals to solve the equations
x 2 8 0 and 3y 2 11 0.
c) Assuming a and b are positive real numbers, solve
the equations x 2 a 0 and ax 2 b 0.
MORE OPERATIONS WITH RADICALS
In this section you will continue studying operations with radicals. We learn to
rationalize some denominators that are different from those rationalized in
Section 9.2.
Dividing Radicals
In Section 9.3 you learned how to add, subtract, and multiply radical expressions.
To divide two radical expressions, simply write the quotient as a ratio and then
simplify, as we did in Section 9.2. In general, we have
n
n
a b n
a
n
b
ab ,
n
provided that all expressions represent real numbers. Note that the quotient rule is
applied only to radicals that have the same index.