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This is “Multiplying and Dividing Radical Expressions”, section 8.4 from the book Beginning Algebra (v. 1.0). For details on it (including licensing), click here.
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8.4 Multiplying and Dividing Radical Expressions
Learning Objectives
1.Multiply radical expressions.
2.Divide radical expressions.
3.Rationalize the denominator.
Multiplying Radical Expressions
When multiplying radical expressions with the same index, we use the product rule for radicals. If a and b represent positive real numbers,
Example 1: Multiply: 2⋅6.
Solution: This problem is a product of two square roots. Apply the product rule for radicals and then simplify.
Answer: 23
Example 2: Multiply: 93⋅63.
Solution: This problem is a product of cube roots. Apply the product rule for radicals and then simplify.
Answer: 3 23
Often there will be coefficients in front of the radicals.
Example 3: Multiply: 23⋅52.
Solution: Using the product rule for radicals and the fact that multiplication is commutative, we can multiply the coefficients and the radicands as follows.
Typically, the first step involving the application of the commutative property is not shown.
Answer: 106
Example 4: Multiply: −2 5x3⋅3 25x23.
Solution:
Answer: −30x
Use the distributive property when multiplying rational expressions with more than one term.
Example 5: Multiply: 43(23−36).
Solution: Apply the distributive property and multiply each term by 43.
Answer: 24−362
Example 6: Multiply: 4x23(2x3−5 4x23).
Solution: Apply the distributive property and then simplify the result.
Answer: 2x−10x⋅2x3
The process for multiplying radical expressions with multiple terms is the same process used when multiplying polynomials. Apply the distributive property, simplify each radical, and then combine like terms.
Example 7: Multiply: (5+2)(5−4).
Solution: Begin by applying the distributive property.
Answer: −3−25
Example 8: Multiply: (3x−y)2.
Solution:
Answer: 9x−6xy+y
Try this! Multiply: (23+52)(3−26).
Answer: 6−122+56−203
Video Solution
(click to see video)
The expressions (a+b) and (a−b) are called conjugatesThe factors (a+b) and (a−b) are conjugates.. When multiplying conjugates, the sum of the products of the inner and outer terms results in 0.
Example 9: Multiply: (2+5)(2−5).
Solution: Apply the distributive property and then combine like terms.
Answer: −3
It is important to note that when multiplying conjugate radical expressions, we obtain a rational expression. This is true in general and is often used in our study of algebra.
Therefore, for nonnegative real numbers a and b, we have the following property:
Dividing Radical Expressions (Rationalizing the Denominator)
To divide radical expressions with the same index, we use the quotient rule for radicals. If a and b represent nonnegative numbers, where b≠0, then we have
Example 10: Divide: 8010.
Solution: In this case, we can see that 10 and 80 have common factors. If we apply the quotient rule for radicals and write it as a single square root, we will be able to reduce the fractional radicand.
Answer: 22
Example 11: Divide: 16x5y42xy.
Solution:
Answer: 2x2y2y
Example 12: Divide: 54a3b5316a2b23.
Solution:
Answer: 3b⋅a32
When the divisor of a radical expression contains a radical, it is a common practice to find an equivalent expression where the denominator is a rational number. Finding such an equivalent expression is called rationalizing the denominatorThe process of determining an equivalent radical expression with a rational denominator..
To do this, multiply the fraction by a special form of 1 so that the radicand in the denominator can be written with a power that matches the index. After doing this, simplify and eliminate the radical in the denominator. For example,
Remember, to obtain an equivalent expression, you must multiply the numerator and denominator by the exact same nonzero factor.
Example 13: Rationalize the denominator: 32.
Solution: The goal is to find an equivalent expression without a radical in the denominator. In this example, multiply by 1 in the form 22.
Answer: 62
Example 14: Rationalize the denominator: 123x.
Solution: The radicand in the denominator determines the factors that you need to use to rationalize it. In this example, multiply by 1 in the form 3x3x.
Answer: 3x6x
Typically, we will find the need to reduce, or cancel, after rationalizing the denominator.
Example 15: Rationalize the denominator: 525ab.
Solution: In this example, we will multiply by 1 in the form 5ab5ab.
Notice that a and b do not cancel in this example. Do not cancel factors inside a radical with those that are outside.
Answer: 10abab
Try this! Rationalize the denominator: 4a3b.
Answer: 23ab3b
Video Solution
(click to see video)
Up to this point, we have seen that multiplying a numerator and a denominator by a square root with the exact same radicand results in a rational denominator. In general, this is true only when the denominator contains a square root. However, this is not the case for a cube root. For example,
Note that multiplying by the same factor in the denominator does not rationalize it. In this case, if we multiply by 1 in the form of x23x23, then we can write the radicand in the denominator as a power of 3. Simplifying the result then yields a rationalized denominator. For example,
Therefore, to rationalize the denominator of radical expressions with one radical term in the denominator, begin by factoring the radicand of the denominator. The factors of this radicand and the index determine what we should multiply by. Multiply numerator and denominator by the nth root of factors that produce nth powers of all the factors in the radicand of the denominator.
Example 16: Rationalize the denominator: 1253.
Solution: The radical in the denominator is equivalent to 523. To rationalize the denominator, it should be 533. To obtain this, we need one more factor of 5. Therefore, multiply by 1 in the form of 5353.
Answer: 535
Example 17: Rationalize the denominator: 27a2b23.
Solution: In this example, we will multiply by 1 in the form 22b322b3.
Answer: 34ab32b
Example 18: Rationalize the denominator: 1 4x35.
Solution: In this example, we will multiply by 1 in the form 23x2523x25.
Answer: 8x252x
When two terms involving square roots appear in the denominator, we can rationalize it using a very special technique. This technique involves multiplying the numerator and the denominator of the fraction by the conjugate of the denominator. Recall that multiplying a radical expression by its conjugate produces a rational number.
Example 19: Rationalize the denominator: 13−2.
Solution: In this example, the conjugate of the denominator is 3+2. Therefore, multiply by 1 in the form (3+2)(3+2).
Answer: 3+2
Notice that the terms involving the square root in the denominator are eliminated by multiplying by the conjugate. We can use the property (a+b)(a−b)=a−b to expedite the process of multiplying the expressions in the denominator.
Example 20: Rationalize the denominator: 2−62+6.
Solution: Multiply by 1 in the form 2−62−6.
Answer: −2+3
Example 21: Rationalize the denominator: x+yx−y.
Solution: In this example, we will multiply by 1 in the form x−yx−y.
Answer: x−2xy+yx−y
Try this! Rationalize the denominator: 35+525−3.
Answer: 195+4511
Video Solution
(click to see video)
Key Takeaways
•To multiply two single-
•Apply the distributive property when multiplying radical expressions with multiple terms. Then simplify and combine all like radicals.
•Multiplying a two-
•It is common practice to write radical expressions without radicals in the denominator. The process of finding such an equivalent expression is called rationalizing the denominator.
•If an expression has one term in the denominator involving a radical, then rationalize it by multiplying numerator and denominator by the nth root of factors of the radicand so that their powers equal the index.
•If a radical expression has two terms in the denominator involving square roots, then rationalize it by multiplying the numerator and denominator by its conjugate.
Topic Exercises
Part A: Multiplying Radical Expressions
Multiply. (Assume all variables are nonnegative.)
1. 3⋅5
2. 7⋅3
3. 2⋅6
4. 5⋅15
5. 7⋅7
6. 12⋅12
7. 25⋅710
8. 315⋅26
9. (25)2
10. (62)2
11. 2x⋅2x
12. 5y⋅5y
13. 3a⋅12
14. 3a⋅2a
15. 42x⋅36x
16. 510y⋅22y
17. 53⋅253
18. 43⋅23
19. 43⋅103
20. 183⋅63
21. (5 93)(2 63)
22. (2 43)(3 43)
23. (2 23)3
24. (3 43)3
25. 3a23⋅9a3
26. 7b3⋅49b23
27. 6x23⋅4x23
28. 12y3⋅9y23
29. 20x2y3⋅10x2y23
30. 63xy3⋅12x4y23
31. 5(3−5)
32. 2(3−2)
33. 37(27−3)
34. 25(6−310)
35. 6(3−2)
36. 15(5+3)
37. x(x+xy)
38. y(xy+y)
39. 2ab(14a−210b)
40. 6ab(52a−3b)
41. (2−5)(3+7)
42. (3+2)(5−7)
43. (23−4)(36+1)
44. (5−26)(7−23)
45. (5−3)2
46. (7−2)2
47. (23+2)(23−2)
48. (2+37)(2−37)
49. (a−2b)2
50. (ab+1)2
51. What are the perimeter and area of a rectangle with length of 53 centimeters and width of 32 centimeters?
52. What are the perimeter and area of a rectangle with length of 26 centimeters and width of 3 centimeters?
53. If the base of a triangle measures 62 meters and the height measures 32 meters, then what is the area?
54. If the base of a triangle measures 63 meters and the height measures 36 meters, then what is the area?
Part B: Dividing Radical Expressions
Divide.
55. 753
56. 36010
57. 7275
58. 9098
59. 90x52x
60. 96y33y
61. 162x7y52xy
62. 363x4y93xy
63. 16a5b232a2b23
64. 192a2b732a2b23
Rationalize the denominator.
65. 15
66. 16
67. 23
68. 37
69. 5210
70. 356
71. 3−53
72. 6−22
73. 17x
74. 13y
75. a5ab
76. 3b223ab
77. 2363
78. 1473
79. 14x3
80. 13y23
81. 9x⋅239xy23
82. 5y2⋅x35x2y3
83. 3a2 3a2b23
84. 25n3 25m2n3
85. 327x2y5
86. 216xy25
87. ab9a3b5
88. abcab2c35
89. 310−3
90. 26−2
91. 15+3
92. 17−2
93. 33+6
94. 55+15
95. 105−35
96. −224−32
97. 3+53−5
98. 10−210+2
99. 23−3243+2
100. 65+225−2
101. x+yx−y
102. x−yx+y
103. a−ba+b
104. ab+2ab−2
105. x5−2x
106. 1x−y
Part C: Discussion
107. Research and discuss some of the reasons why it is a common practice to rationalize the denominator.
108. Explain in your own words how to rationalize the denominator.
Answers
1: 15
3: 23
5: 7
7: 702
9: 20
11: 2x
13: 6a
15: 24x3
17: 5
19: 2 53
21: 30 23
23: 16
25: 3a
27: 2x⋅3x3
29: 2xy⋅25x3
31: 35−5
33: 42−321
35: 32−23
37: x+xy
39: 2a7b−4b5a
41: 6+14−15−35
43: 182+23−126−4
45: 8−215
47: 10
49: a−22ab+2b
51: Perimeter: (103+62) centimeters; area: 156 square centimeters
53: 18 square meters
55: 5
57: 265
59: 3x25
61: 9x3y2
63: 2a
65: 55
67: 63
69: 104
71: 3−153
73: 7x7x
75: ab5b
77: 633
79: 2x232x
81: 3 6x2y3y
83: 9ab32b
85: 9x3y45xy
87: 27a2b453
89: 310+9
91: 5−32
93: −1+2
95: −5−352
97: −4−15
99: 15−7623
101: x2+2xy+yx2−y
103: a−2ab+ba−b
105: 5x+2x25−4x
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