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This is “Simplifying Radical Expressions”, section 8.2 from the book Beginning Algebra (v. 1.0). For details on it (including licensing), click here.
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8.2 Simplifying Radical Expressions
Learning Objectives
1.Simplify radical expressions using the product and quotient rule for radicals.
2.Use formulas involving radicals.
3.Evaluate given square root and cube root functions.
Simplifying Radical Expressions
An algebraic expression that contains radicals is called a radical expressionAn algebraic expression that contains radicals.. We use the product and quotient rules to simplify them.
Example 1: Simplify: 8y33.
Solution: Use the fact that ann=a when n is odd.
Answer: 2y
Example 2: Simplify: 9x2.
Solution: The square root has index 2; use the fact that ann=|a| when n is even.
Since x is a variable, it may represent a negative number. Thus we need to ensure that the result is positive by including the absolute value operator.
Answer: 3|x|
Important Note
Typically, at this point beginning algebra texts note that all variables are assumed to be positive. If this is the case, then x in the previous example is positive and the absolute value operator is not needed. The example can be simplified as follows:
9x2=32x2 =32⋅x2=3x
In this section, we will assume that all variables are positive. This allows us to focus on calculating nth roots without the technicalities associated with the principal nth root problem. For this reason, we will use the following property for the rest of the section:
ann=a, if a≥0 nth root
When simplifying radical expressions, look for factors with powers that match the index.
Example 3: Simplify: 18x3y4.
Solution: Begin by determining the square factors of 18, x3, and y4.
Make these substitutions and then apply the product rule for radicals and simplify.
Answer: 3xy22x
Example 4: Simplify: 4a5b6.
Solution: Begin by determining the square factors of 4, a5, and b6.
Make these substitutions and then apply the product rule for radicals and simplify.
Answer: 2a2ab3
Example 5: Simplify: 80x5y73.
Solution: Begin by determining the cubic factors of 80, x5, and y7.
Make these substitutions and then apply the product rule for radicals and simplify.
Answer: 2xy2⋅10x2y3
Example 6: Simplify 9x6y3z93.
Solution: The coefficient 9=32 and thus does not have any perfect cube factors. It will be left as the only remaining radicand because all of the other factors are cubes, as illustrated below:
Replace the variables with these equivalents, apply the product and quotient rule for radicals, and then simplify.
Answer: x2⋅93yz3
Example 7: Simplify: 81a4b54.
Solution: Determine all factors that can be written as perfect powers of 4. Here it is important to see that b5=b4⋅b. Hence the factor b will be left inside the radical.
Answer: 3ab⋅b4
Example 8: Simplify: −32x3y6z55.
Solution: Notice that the variable factor x cannot be written as a power of 5 and thus will be left inside the radical. In addition, for y6=y5⋅y; the factor y will be left inside the radical as well.
Answer: −2yz⋅x3y5
Try this! Simplify: 192x6y7z12. (Assume all variables are positive.)
Answer: 8x3y3z63y
Video Solution
(click to see video)
Tip
To easily simplify an nth root, we can divide the powers by the index.
a6=a3, which is a6÷2=a3b63=b2, which is b6÷3=b2c66=c , which is c6÷6=c1
If the index does not divide into the power evenly, then we can use the quotient and remainder to simplify. For example,
a5=a2⋅a, which is a5÷2=a2 r 1b53=b⋅b23, which is b5÷3=b1 r 2c145=c2⋅c45, which is c14÷5=c2 r 4
The quotient is the exponent of the factor outside of the radical, and the remainder is the exponent of the factor left inside the radical.
Formulas Involving Radicals
We next review the distance formula. Given two points (x1, y1) and (x2, y2),
The distance, d, between them is given by the following formula:
Distance formulaGiven two points (x1, y1) and (x2, y2), calculate the distance d between them using the formula d = ( x 2− x 1)2+( y 2− y 1)2.:
d=( x 2− x 1)2+( y 2− y 1)2
Recall that this formula was derived from the Pythagorean theorem.
Example 9: Calculate the distance between (−4, 7) and (2, 1).
Solution: Use the distance formula with the following points.
It is a good practice to include the formula in its general form before substituting values for the variables; this improves readability and reduces the probability of making errors.
Answer: 62 units
Example 10: The period, T, of a pendulum in seconds is given by the formula
where L represents the length of the pendulum in feet. If the length of a pendulum measures 6 feet, then calculate the period rounded off to the nearest tenth of a second.
Solution: Substitute 6 for L and then simplify.
Answer: The period is approximately 2.7 seconds.
Square Root and Cube Root Functions
We begin with the square root functionThe function f(x)=x.:
We know that the square root is not a real number when the radicand x is negative. Therefore, we conclude that the domain consists of all real numbers greater than or equal to 0. Here we choose 0 and some positive values for x, calculate the corresponding y-
After plotting the points, we can then sketch the graph of the square root function.
Example 11: Given the function f(x)=x+2, find f(−2), f(2), and f(6).
Solution: Replace x with each of the given values.
Answer: f(−2)=0, f(2)=2, and f(6)=22
Next, consider the cube root functionThe function f(x)=x3.:
Since the cube root could be either negative or positive, we conclude that the domain consists of all real numbers. For completeness, choose some positive and negative values for x, as well as 0, and then calculate the corresponding y-
Plot the points and sketch the graph of the cube root function.
Example 12: Given the function g(x)=x−13, find g(−7), g(0), and g(55).
Solution: Replace x with each of the given values.
Answer: g(−7)=−2, g(0)=−1, and g(55)=3 23
Key Takeaways
•In beginning algebra, we typically assume that all variable expressions within the radical are positive. This allows us to focus on simplifying radicals without the technical issues associated with the principal nth root.
•To simplify radical expressions, look for factors of the radicand with powers that match the index. If found, they can be simplified by applying the product and quotient rules for radicals, as well as the property ann=a, where a is positive.
Topic Exercises
Part A: Simplifying Radical Expressions
Simplify. (Assume all variables represent positive numbers.)
1. 36a2
2. 121b2
3. x2y2
4. 25x2y2z2
5. 180x3
6. 150y3
7. 49a3b2
8. 4a4b3c
9. 45x5y3
10. 50x6y4
11. 64r2s6t5
12. 144r8s6t2
13. (x+1)2
14. (2x+3)2
15. 4(3x−1)2
16. 9(2x+3)2
17. 9x325y2
18. 4x59y4
19. m736n4
20. 147m9n6
21. 2r2s525t4
22. 36r5s2t6
23. 27a33
24. 125b33
25. 250x4y33
26. 162a3b53
27. 64x3y6z93
28. 216x12y33
29. 8x3y43
30. 27x5y33
31. a4b5c63
32. a7b5c33
33. 8x427y33
34. x5125y63
35. 360r5s12t133
36. 540r3s2t93
37. 81x44
38. x4y44
39. 16x4y84
40. 81x12y44
41. a4b5c64
42. 54a6c84
43. 128x64
44. 243y74
45. 32m10n55
46. 37m9n105
47. −34x2
48. 79y2
49. −5x4x2y
50. −3y16x3y2
51. 12aba5b3
52. 6a2b9a7b2
53. 2x⋅8x63
54. −5x2⋅27x33
55. 2ab⋅−8a4b53
56. 5a2b⋅−27a3b33
Rewrite the following as a radical expression with coefficient 1.
57. 52x
58. 23y
59. 2x3
60. 3y2
61. ab10a
62. 2ab2a
63. m2nmn
64. 2m2n33n
65. 5 2x3
66. 3 5y3
67. 2x⋅33
68. 3y⋅23
Assume that the variable could represent any real number and then simplify.
69. 4x2
70. 25y2
71. 8y33
72. 125a33
73. 64x44
74. 81y44
75. 36a4
76. 100a8
77. 4a6
78. a10
79. 18a4b5
80. 48a5b3
81. 128x6y86
82. a6b7c86
Part B: Formulas Involving Radicals
The y-
83. y=x+4−1
84. y=x+1−3
85. y=x−13+2
86. y=x+13−3
Use the distance formula to calculate the distance between the given two points.
87. (5, −7) and (3, −8)
88. (−9, 7) and (−8, 4)
89. (−3, −4) and (3, −6)
90. (−5, −2) and (1, −6)
91. (−1, 1) and (−4, 10)
92. (8, −3) and (2, −12)
Factor the radicand and then simplify. (Assume that all expressions are positive.)
93. x2−6x+9
94. x2−10x+25
95. 4x2+12x+9
96. 9x2+6x+1
97. The speed of a vehicle before the brakes were applied can be estimated by the length of the skid marks left on the road. On dry pavement, the speed, v, in miles per hour can be estimated by the formula v=5d, where d represents the length of the skid marks in feet. Estimate the speed of a vehicle before applying the brakes on dry pavement if the skid marks left behind measure 36 feet.
98. The radius, r, of a sphere can be calculated using the formula r=6π2V32π, where V represents the sphere’s volume. What is the radius of a sphere if the volume is 36π cubic centimeters?
The period, T, of a pendulum in seconds is given by the formula
T=2πL32
where L represents the length in feet. Calculate the period, given the following lengths. Give the exact value and the approximate value rounded off to the nearest tenth of a second.
99. 8 feet
100. 32 feet
101. 1/2 foot
102. 1/8 foot
The time, t, in seconds that an object is in free fall is given by the formula t=s4
where s represents the distance it has fallen in feet. Calculate the time it takes an object to fall, given the following distances. Give the exact value and the approximate value rounded off to the nearest tenth of a second.
103. 48 feet
104. 80 feet
105. 192 feet
106. 288 feet
Part C: Radical Functions
Given the function, calculate the following.
107. f(x)=x−1, find f(1), f(2), and f(5)
108. f(x)=x+5, find f(−5), f(−1), and f(20)
109. f(x)=x+3, find f(0), f(1), and f(16)
110. f(x)=x−5, find f(0), f(1), and f(25)
111. g(x)=x3, find g(−1), g(0), and g(1)
112. g(x)=x+73, find g(−15), g(−7), and g(20)
113. g(x)=x3−2, find g(−1), g(0), and g(8)
114. g(x)=x−13+2, find g(0), g(2), and g(9)
For each function, fill in the table.
115. f(x)=x+1
116. f(x)=x−2
117. f(x)=x3+1
118. f(x)=x+23
Part D: Discussion Board
119. Give a value for x such that x2≠x. Explain why it is important to assume that the variables represent positive numbers.
120. Research and discuss the accomplishments of Christoph Rudolff. What is he credited for?
121. Research and discuss the methods used for calculating square roots before the common use of electronic calculators.
122. What is a surd, and where does the word come from?
Answers
1: 6a
3: xy
5: 6x5x
7: 7aba
9: 3x2y5xy
11: 8rs3t2t
13: x+1
15: 2(3x−1)
17: 3xx5y
19: m3m6n2
21: rs22s5t2
23: 3a
25: 5xy⋅2x3
27: 4xy2z3
29: 2xy⋅y3
31: abc2⋅ab23
33: 2x⋅x33y
35: 2rs4t4⋅45r2t3
37: 3x
39: 2xy2
41: abc⋅bc24
43: 2x⋅8x24
45: 2m2n
47: −6x
49: −10x2y
51: 12a3b2ab
53: 4x3
55: −4a2b2⋅ab23
57: 50x
59: 12x2
61: 10a3b2
63: m5n3
65: 250x3
67: 24x33
69: 2|x|
71: 2y
73: 2|x|
75: 6a2
77: 2|a3|
79: 3a2b22b
81: 2|xy|⋅2y26
83: (0, 1)
85: (0, 1)
87: 5
89: 210
91: 310
93: x−3
95: 2x+3
97: 30 miles per hour
99: π≈3.1 seconds
101: π/4≈0.8 seconds
103: 3≈1.7 seconds
105: 23≈3.5 seconds
107: f(1)=0, f(2)=1, and f(5)=2
109: f(0)=3, f(1)=4, and f(16)=7
111: g(−1)=−1, g(0)=0, and g(1)=1
113: g(−1)=−3, g(0)=−2, and g(8)=0
115:
117:
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