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Chapter 8 Radical Expressions and Equations
8.1 Radicals
Learning Objectives
1.Find square roots.
2.Find cube roots.
3.Find nth roots.
4.Simplify expressions using the product and quotient rules for radicals.
Square Roots
The square rootThe number that, when multiplied by itself, yields the original number. of a number is that number that when multiplied by itself yields the original number. For example, 4 is a square root of 16, because 4 2 =16. Since ( −4 ) 2 =16, we can say that −4 is a square root of 16 as well. Every positive real number has two square roots, one positive and one negative. For this reason, we use the radical sign to denote the principal (nonnegative) square rootThe positive square root of a real number, denoted with the symbol . and a negative sign in front of the radical − to denote the negative square root.
Zero is the only real number with one square root.
If the radicandThe expression a within a radical sign, an., the number inside the radical sign, is nonnegative and can be factored as the square of another nonnegative number, then the square root of the number is apparent. In this case, we have the following property:
Example 1: Find the square root.
a. 36
b. 144
c. 0.04
d. 19
Solution:
a. 36=62=6
b. 144=122=12
c. 0.04=(0.2)2=0.2
d. 19=( 1 3)2=13
Example 2: Find the negative square root.
a. −4
b. −1
Solution:
a. −4=−22=−2
b. −1=−12=−1
The radicand may not always be a perfect square. If a positive integer is not a perfect square, then its square root will be irrational. For example, 2 is an irrational number and can be approximated on most calculators using the square root button.
Next, consider the square root of a negative number. To determine the square root of −9, you must find a number that when squared results in −9:
However, any real number squared always results in a positive number:
The square root of a negative number is currently left undefined. For now, we will state that −9 is not a real a number.
Cube Roots
The cube rootThe number that, when used as a factor with itself three times, yields the original number; it is denoted with the symbol 3. of a number is that number that when multiplied by itself three times yields the original number. Furthermore, we denote a cube root using the symbol 3, where 3 is called the indexThe positive integer n in the notation n that is used to indicate an nth root.. For example,
The product of three equal factors will be positive if the factor is positive and negative if the factor is negative. For this reason, any real number will have only one real cube root. Hence the technicalities associated with the principal root do not apply. For example,
In general, given any real number a, we have the following property:
When simplifying cube roots, look for factors that are perfect cubes.
Example 3: Find the cube root.
a. 273
b. 643
c. 03
d. 183
Solution:
a. 273=333=3
b. 643=433=4
c. 03=033=0
d. 183=( 1 2)33=12
Example 4: Find the cube root.
a. −83
b. −13
c. −1273
Solution:
a. −83=(−2)33=−2
b. −13=(−1)33=−1
c. −1273=(− 1 3)33=−13
It may be the case that the radicand is not a perfect cube. If an integer is not a perfect cube, then its cube root will be irrational. For example, 23 is an irrational number which can be approximated on most calculators using the root button. Depending on the calculator, we typically type in the index prior to pushing the button and then the radicand as follows:
Therefore, we have
nth Roots
For any integer n≥2, we define the nth rootThe number that, when raised to the nth power, yields the original number. of a positive real number as that number that when raised to the nth power yields the original number. Given any nonnegative real number a, we have the following property:
Here n is called the index and an is called the radicand. Furthermore, we can refer to the entire expression an as a radicalUsed when referring to an expression of the form an.. When the index is an integer greater than 3, we say “fourth root”, “fifth root”, and so on. The nth root of any number is apparent if we can write the radicand with an exponent equal to the index.
Example 5: Find the nth root.
a. 814
b. 325
c. 17
d. 1164
Solution:
a. 814=344=3
b. 325=255=2
c. 17=177=1
d. 1164=( 1 2)44=12
If the index is n=2, then the radical indicates a square root and it is customary to write the radical without the index, as illustrated below:
We have already taken care to define the principal square root of a number. At this point, we extend this idea to nth roots when n is even. For example, 3 is a fourth root of 81, because 34=81. And since (−3)4=81, we can say that −3 is a fourth root of 81 as well. Hence we use the radical sign n to denote the principal (nonnegative) nth rootThe positive nth root when n is even. when n is even. In this case, for any real number a, we use the following property:
For example,
The negative nth root, when n is even, will be denoted using a negative sign in front of the radical − n.
We have seen that the square root of a negative number is not real because any real number, when squared, will result in a positive number. In fact, a similar problem arises for any even index:
Here the fourth root of −81 is not a real number because the fourth power of any real number is always positive.
Example 6: Simplify.
a. −164
b. −164
Solution:
a. The radicand is negative and the index is even. Therefore, there is no real number that when raised to the fourth power is −16.
b. Here the radicand is positive. Furthermore, 16=24, and we can simplify as follows:
When n is odd, the same problems do not occur. The product of an odd number of positive factors is positive and the product of an odd number of negative factors is negative. Hence when the index n is odd, there is only one real nth root for any real number a. And we have the following property:
Example 7: Find the nth root.
a. −325
b. −17
Solution:
a. −325=(−2)55=−2
b. −17=(−1)77=−1
Try this! Find the fourth root: 6254.
Answer: 5
Video Solution
(click to see video)
Summary: When n is odd, the nth root is positive or negative depending on the sign of the radicand.
When n is even, the nth root is positive or not real depending on the sign of the radicand.
Simplifying Using the Product and Quotient Rule for Radicals
It will not always be the case that the radicand is a perfect power of the given index. If not, we use the following two properties to simplify them. If a and b represent positive real numbers, then we have
Product rule for radicalsa⋅bn=an⋅bn, where a and b represent positive real numbers.:
a⋅bn=an⋅bn
Quotient rule for radicalsabn=anbn, where a and b represent positive real numbers.:
abn=anbn
A radical is simplifiedA radical where the radicand does not consist of any factor that can be written as a perfect power of the index. if it does not contain any factor that can be written as a perfect power of the index.
Example 8: Simplify: 12.
Solution: Here 12 can be written as 4 ⋅ 3, where 4 is a perfect square.
We can verify our answer on a calculator:
Also, it is worth noting that
Answer: 23
Example 9: Simplify: 135.
Solution: Begin by finding the largest perfect square factor of 135.
Therefore,
Answer: 315
Example 10: Simplify: 50121.
Solution: Begin by finding the prime factorizations of both 50 and 121. This will enable us to easily determine the largest perfect square factors.
Therefore,
Answer: 5211
Example 11: Simplify: 1623.
Solution: Use the prime factorization of 162 to find the largest perfect cube factor:
Replace the radicand with this factorization and then apply the product rule for radicals.
We can verify our answer on a calculator.
Answer: 3 63
Try this! Simplify: 2 963.
Answer: 4 123
Video Solution
(click to see video)
Example 12: Simplify: −965.
Solution: Here we note that the index is odd and the radicand is negative; hence the result will be negative. We can factor the radicand as follows:
Then simplify:
Answer: −2 35
Example 13: Simplify: −8643.
Solution: In this case, consider the equivalent fraction with −8=(−2)3 in the numerator and then simplify.
Answer: −1/2
Try this! Simplify −1083.
Answer: −3 43
Video Solution
(click to see video)
Key Takeaways
•The square root of a number is that number that when multiplied by itself yields the original number. When the radicand a is positive, a2=a. When the radicand is negative, the result is not a real number.
•The cube root of a number is that number that when used as a factor with itself three times yields the original number. The cube root may be positive or negative depending on the sign of the radicand. Therefore, for any real number a, we have the property a33=a.
•When working with nth roots, n determines the definition that applies. We use ann=a when n is odd and ann=|a| when n is even. When n is even, the negative nth root is denoted with a negative sign in front of the radical sign.
•To simplify square roots, look for the largest perfect square factor of the radicand and then apply the product or quotient rule for radicals.
•To simplify cube roots, look for the largest perfect cube factor of the radicand and then apply the product or quotient rule for radicals.
•To simplify nth roots, look for the factors that have a power that is equal to the index n and then apply the product or quotient rule for radicals. Typically, the process is streamlined if you work with the prime factorization of the radicand.
Topic Exercises
Part A: Radicals
Simplify.
1. 81
2. 100
3. 64
4. 121
5. 0
6. 1
7. 0.25
8. 0.01
9. 1.21
10. 2.25
11. 14
12. 136
13. 2516
14. 925
15. −25
16. −9
17. −36
18. −81
19. −100
20. −1
21. 273
22. 1253
23. 643
24. 83
25. 183
26. 1643
27. 8273
28. 641253
29. 0.0013
30. 1,0003
31. −13
32. −83
33. −273
34. −643
35. −183
36. −27643
37. −8273
38. −11253
39. 814
40. 6254
41. 164
42. 10,0004
43. 325
44. 15
45. 2435
46. 100,0005
47. −164
48. −16
49. −325
50. −15
51. −1
52. −164
53. −5 −273
54. −2 −83
55. 5 −1,0003
56. 3 −2435
57. 10 −164
58. 2 −646
59. 325
60. 64
61. 2 273
62. 8 2435
63. −7 83
64. −4 6254
65. 6 100,0005
66. 5 1287
Part B: Simplifying Radicals
Simplify.
67. 32
68. 250
69. 80
70. 150
71. 160
72. 60
73. 175
74. 216
75. 5112
76. 10135
77. 5049
78. −2120
79. −3162
80. 89
81. 45121
82. 9681
83. 543
84. 243
85. 483
86. 813
87. 403
88. 1203
89. 1623
90. 5003
91. 541253
92. 403433
93. 5 −483
94. 2 −1083
95. 8 964
96. 7 1624
97. 1605
98. 4865
99. 2242435
100. 5325
Simplify. Give the exact answer and the approximate answer rounded to the nearest hundredth.
101. 8
102. 200
103. 45
104. 72
105. 34
106. 59
107. 3225
108. 4849
109. 803
110. 3203
111. 483
112. 2703
Rewrite the following as a radical expression with coefficient 1.
113. 215
114. 37
115. 510
116. 103
117. 2 73
118. 3 63
119. 2 54
120. 3 24
121. The formula for the area A of a square is A=s2. If the area is 18 square units, then what is the length of each side?
122. Calculate the length of a side of a square with an area of 60 square centimeters.
123. The formula for the volume V of a cube is V=s3. If the volume of a cube is 112 cubic units, then what is the length of each side?
124. Calculate the length of a side of a cube with a volume of 54 cubic centimeters.
Part C: Discussion Board
125. Explain why there are two square roots for any nonzero real number.
126. Explain why there is only one cube root for any real number.
127. What is the square root of 1, and what is the cube root of 1? Explain why.
128. Explain why −1 is not a real number and why −13 is a real number.
Answers
1: 9
3: 8
5: 0
7: 0.5
9: 1.1
11: 1/2
13: 5/4
15: Not a real number
17: −6
19: −10
21: 3
23: 4
25: 1/2
27: 2/3
29: 0.1
31: −1
33: −3
35: −1/2
37: −2/3
39: 3
41: 2
43: 2
45: 3
47: −2
49: −2
51: Not a real number
53: 15
55: −50
57: Not a real number
59: 15
61: 6
63: −14
65: 60
67: 42
69: 45
71: 410
73: 57
75: 207
77: 527
79: −272
81: 3511
83: 3 23
85: 2 63
87: 2 53
89: 3 63
91: 3 235
93: −10 63
95: 16 64
97: 2 55
99: 2 753
101: 22≈2.83
103: 35≈6.71
105: 32≈0.87
107: 425≈1.13
109: 2 103≈4.31
111: 2 63≈3.63
113: 60
115: 250
117: 563
119: 804
121: 32 units
123: 2 143 units
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:
8.3 Adding and Subtracting Radical Expressions
Learning Objectives
1.Add and subtract like radicals.
2.Simplify radical expressions involving like radicals.
Adding and Subtracting Radical Expressions
Adding and subtracting radical expressions is similar to adding and subtracting like terms. Radicals are considered to be like radicalsRadicals that share the same index and radicand., or similar radicalsTerm used when referring to like radicals., when they share the same index and radicand. For example, the terms 35 and 45 contain like radicals and can be added using the distributive property as follows:
Typically, we do not show the step involving the distributive property and simply write
When adding terms with like radicals, add only the coefficients; the radical part remains the same.
Example 1: Add: 32+22.
Solution: The terms contain like radicals; therefore, add the coefficients.
Answer: 52
Subtraction is performed in a similar manner.
Example 2: Subtract: 27−37.
Solution:
Answer: −7
If the radicand and the index are not exactly the same, then the radicals are not similar and we cannot combine them.
Example 3: Simplify: 105+62−95−72.
Solution:
We cannot simplify any further because 5 and 2 are not like radicals; the radicands are not the same.
Answer: 5−2
Caution
It is important to point out that 5−2≠5−2. We can verify this by calculating the value of each side with a calculator.
In general, note that an±bn≠a±bn.
Example 4: Simplify: 3 63+26−63−36.
Solution:
We cannot simplify any further because 63 and 6 are not like radicals; the indices are not the same.
Answer: 2 63−6
Often we will have to simplify before we can identify the like radicals within the terms.
Example 5: Subtract: 12−48.
Solution: At first glance, the radicals do not appear to be similar. However, after simplifying completely, we will see that we can combine them.
Answer: −23
Example 6: Simplify: 20+27−35−212.
Solution:
Answer: −5−3
Try this! Subtract: 250−68.
Answer: −22
Video Solution
(click to see video)
Next, we work with radical expressions involving variables. In this section, assume all radicands containing variable expressions are not negative.
Example 7: Simplify: −6 2x3−3x3+7 2x3.
Solution:
We cannot combine any further because the remaining radical expressions do not share the same radicand; they are not like radicals. Note that 2x3−3x3≠2x−3x3.
Answer: 2x3−3x3
We will often find the need to subtract a radical expression with multiple terms. If this is the case, remember to apply the distributive property before combining like terms.
Example 8: Simplify: (9x−2y)−(10x+7y).
Solution:
Answer: −x−9y
Until we simplify, it is often unclear which terms involving radicals are similar.
Example 9: Simplify: 5 2y3−(54y3−163).
Solution:
Answer: 2 2y3+2 23
Example 10: Simplify: 2a125a2b−a280b+420a4b.
Solution:
Answer: 14a25b
Try this! Simplify: 45x3−(20x3−80x).
Answer: x5x+45x
Video Solution
(click to see video)
Tip
Take careful note of the differences between products and sums within a radical.
Products
Sums
x2y2=xyx3y33=xy x2+y2≠x+yx3+y33≠x+y
The property a⋅bn=an⋅bn says that we can simplify radicals when the operation in the radicand is multiplication. There is no corresponding property for addition.
Key Takeaways
•Add and subtract terms that contain like radicals just as you do like terms. If the index and radicand are exactly the same, then the radicals are similar and can be combined. This involves adding or subtracting only the coefficients; the radical part remains the same.
•Simplify each radical completely before combining like terms.
Topic Exercises
Part A: Adding and Subtracting Like Radicals
Simplify.
1. 93+53
2. 126+36
3. 45−75
4. 310−810
5. 6−46+26
6. 510−1510−210
7. 137−62−57+52
8. 1013−1215+513−1815
9. 65−(43−35)
10. −122−(66+2)
11. (25−310)−(10+35)
12. (−83+615)−(3−15)
13. 4 63−3 53+6 63
14. 103+5 103−4 103
15. (7 93−4 33)−(93−3 33)
16. (−8 53+253)−(2 53+6 253)
Simplify. (Assume all radicands containing variable expressions are positive.)
17. 9x+7x
18. −8y+4y
19. 7xy−3xy+xy
20. 10y2x−12y2x−2y2x
21. 2ab−5a+6ab−10a
22. −3xy+6y−4xy−7y
23. 5xy−(3xy−7xy)
24. −8ab−(2ab−4ab)
25. (32x−3x)−(2x−73x)
26. (y−42y)−(y−52y)
27. 5 x3−12 x3
28. −2 y3−3 y3
29. a⋅3b5+4a⋅3b5−a⋅3b5
30. −8 ab4+3 ab4−2 ab4
31. 62a−4 2a3+72a−2a3
32. 4 3a5+3a3−9 3a5+3a3
33. (4xy4−xy3)−(2 4xy4−xy3)
34. (5 6y6−5y)−(2 6y6+3y)
Part B: Adding and Subtracting Rational Expressions
Simplify.
35. 75−12
36. 24−54
37. 32+27−8
38. 20+48−45
39. 28−27+63−12
40. 90+24−40−54
41. 45−80+245−5
42. 108+48−75−3
43. 42−(27−72)
44. −35−(20−50)
45. 163−543
46. 813−243
47. 1353+403−53
48. 1083−323−43
49. 227−212
50. 350−432
51. 3243−218−48
52. 6216−224−296
53. 218−375−298+448
54. 245−12+220−108
55. (2363−396)−(712−254)
56. (2288+3360)−(272−740)
57. 3 543+5 2503−4 163
58. 4 1623−2 3843−3 7503
Simplify. (Assume all radicands containing variable expressions are positive.)
59. 81b+4b
60. 100a+a
61. 9a2b−36a2b
62. 50a2−18a2
63. 49x−9y+x−4y
64. 9x+64y−25x−y
65. 78x−(316y−218x)
66. 264y−(332y−81y)
67. 29m2n−5m9n+m2n
68. 418n2m−2n8m+n2m
69. 4x2y−9xy2−16x2y+y2x
70. 32x2y2+12x2y−18x2y2−27x2y
71. (9x2y−16y)−(49x2y−4y)
72. (72x2y2−18x2y)−(50x2y2+x2y)
73. 12m4n−m75m2n+227m4n
74. 5n27mn2+212mn4−n3mn2
75. 227a3b−a48ab−a144a3b
76. 298a4b−2a162a2b+a200b
77. 125a3−27a3
78. 1000a23−64a23
79. 2x⋅54x3−2 16x43+5 2x43
80. x⋅54x33−250x63+x2⋅23
81. 16y24+81y24
82. 32y45−y45
83. 32a34−162a34+5 2a34
84. 80a4b4+5a4b4−a⋅5b4
85. 27x33+8x3−125x33
86. 24x3−128x3−81x3
87. 27x4y3−8xy33+x⋅64xy3−y⋅x3
88. 125xy33+8x3y3−216xy33+10x⋅y3
89. (162x4y3−250x4y23)−(2x4y23−384x4y3)
90. (32x2y65−243x6y25)−(x2y65−x⋅xy25)
Part C: Discussion Board
91. Choose values for x and y and use a calculator to show that x+y≠x+y.
92. Choose values for x and y and use a calculator to show that x2+y2≠x+y.
Answers
1: 143
3: −25
5: −6
7: 87−2
9: 95−43
11: −5−410
13: 10 63−3 53
15: 6 93−33
17: 16x
19: 5xy
21: 8ab−15a
23: 9xy
25: 22x+63x
27: −7 x3
29: 4a⋅3b5
31: 132a−5 2a3
33: −4xy4
35: 33
37: 22+33
39: 57−53
41: 55
43: 102−33
45: −23
47: 4 53
49: 23
51: 233−62
53: −82+3
55: 83−66
57: 26 23
59: 11b
61: −3ab
63: 8x−5y
65: 202x−12y
67: −8mn
69: −2xy−2yx
71: −4xy
73: 3m23n
75: 2a3ab−12a2ab
77: 2 a3
79: 7x⋅2x3
81: 5 y24
83: 4 2a34
85: −2x+2 x3
87: 7x⋅xy3−3y⋅x3
89: 7x⋅6xy3−6x⋅2xy23
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
8.5 Rational Exponents
Learning Objectives
1.Write expressions with rational exponents in radical form.
2.Write radical expressions with rational exponents.
3.Perform operations and simplify expressions with rational exponents.
4.Perform operations on radicals with different indices.
Definition of Rational Exponents
So far, exponents have been limited to integers. In this section, we will define what rational (or fractional) exponentsThe fractional exponent m/n that indicates a radical with index n and exponent m: am/n=amn. mean and how to work with them. All of the rules for exponents developed up to this point apply. In particular, recall the product rule for exponents. Given any rational numbers m and n, then
For example, if we have an exponent of 12, then the product rule for exponents implies the following:
Here 51/2 is one of two equal factors of 5; hence it is a square root of 5, and we can write
Furthermore, we can see that 21/3 is one of three equal factors of 2.
Therefore, 21/3 is the cube root of 2, and we can write
This is true in general, given any nonzero real number a,
In other words, the denominator of a fractional exponent determines the index of an nth root.
Example 1: Rewrite as a radical.
a. 71/2
b. 71/3
Solution:
a. 71/2=7
b. 71/3=73
Example 2: Rewrite as a radical and then simplify.
a. 811/2
b. 811/4
Solution:
a. 811/2=81=9
b. 811/4=814=344=3
Example 3: Rewrite as a radical and then simplify.
a. (125x3)1/3
b. (−32y10)1/5
Solution:
a.
b.
Next, consider fractional exponents where the numerator is an integer other than 1. For example, consider the following:
This shows that 52/3 is one of three equal factors of 52. In other words, 52/3 is the cube root of 52 and we can write:
In general, given any real number a,
An expression with a rational exponent is equivalent to a radical where the denominator is the index and the numerator is the exponent. Any radical expression can be written with a rational exponent, which we call exponential formAn equivalent expression written using a rational exponent..
Example 4: Rewrite as a radical.
a. 72/5
b. 23/4
Solution:
a. 72/5=725=495
b. 23/4=234=84
Example 5: Rewrite as a radical and then simplify.
a. 82/3
b. (32)3/5
Solution:
a.
b. We can often avoid very large integers by working with their prime factorization.
Given a radical expression, we will be asked to find the equivalent in exponential form. Assume all variables are positive.
Example 6: Rewrite using rational exponents: x23.
Solution: Here the index is 3 and the power is 2. We can write
Answer: x2/3
Example 7: Rewrite using rational exponents: y36.
Solution: Here the index is 6 and the power is 3. We can write
Answer: y1/2
It is important to note that the following are equivalent.
In other words, it does not matter if we apply the power first or the root first. For example, we can apply the power before the root:
Or we can apply the nth root before the power:
The results are the same.
Example 8: Rewrite as a radical and then simplify: (−8)2/3.
Solution: Here the index is 3 and the power is 2. We can write
Answer: 4
Try this! Rewrite as a radical and then simplify: 253/2.
Answer: 125
Video Solution
(click to see video)
Some calculators have a caret button ^. If so, we can calculate approximations for radicals using it and rational exponents. For example, to calculate 2=21/2=2^(1/2)≈1.414, we would type
To calculate 223=22/3=2^(2/3)≈1.587, we would type
Operations Using the Rules of Exponents
In this section, we review all of the rules of exponents, which extend to include rational exponents. If given any rational numbers m and n, then we have
Product rule: xm⋅xn=xm+n
Quotient rule: xmxn=xm−n , x≠0
Power rule: (xm)n=xm⋅n
Power rule for a product: (xy)n=xnyn
Power rule for a quotient: (xy)n=xnyn , y≠0
Negative exponents: x−n=1xn
Zero exponent: x0=1, x≠0
These rules allow us to perform operations with rational exponents.
Example 9: Simplify: 22/3⋅21/6.
Solution:
Answer: 25/6
Example 10: Simplify: x1/2x1/3.
Solution:
Answer: x1/6
Example 11: Simplify: (y3/4)2/3.
Solution:
Answer: y1/2
Example 12: Simplify: (16a4b8)3/4.
Solution:
Answer: 8a3b6
Example 13: Simplify: 25−3/2.
Solution:
Answer: 1/125
Try this! Simplify: (8 a 3/4 b 3)2/3a1/3.
Answer: 4a1/6b2
Video Solution
(click to see video)
Radical Expressions with Different Indices
To apply the product or quotient rule for radicals, the indices of the radicals involved must be the same. If the indices are different, then first rewrite the radicals in exponential form and then apply the rules for exponents.
Example 14: Multiply: 2⋅23.
Solution: In this example, the index of each radical factor is different. Hence the product rule for radicals does not apply. Begin by converting the radicals into an equivalent form using rational exponents. Then apply the product rule for exponents.
Answer: 256
Example 15: Divide: 4325.
Solution: In this example, the index of the radical in the numerator is different from the index of the radical in the denominator. Hence the quotient rule for radicals does not apply. Begin by converting the radicals into an equivalent form using rational exponents and then apply the quotient rule for exponents.
Answer: 2715
Example 16: Simplify: 43 .
Solution: Here the radicand of the square root is a cube root. After rewriting this expression using rational exponents, we will see that the power rule for exponents applies.
Answer: 23
Key Takeaways
•When converting fractional exponents to radicals, use the numerator as the power and the denominator as the index of the radical.
•All the rules of exponents apply to expressions with rational exponents.
Topic Exercises
Part A: Rational Exponents
Express using rational exponents.
1. 6
2. 10
3. 113
4. 24
5. 523
6. 234
7. x5
8. x6
9. x76
10. x45
Express in radical form.
11. 21/2
12. 51/3
13. 72/3
14. 23/5
15. x3/4
16. x5/6
17. x−1/2
18. x−3/4
19. (1x)−1/3
20. (1x)−3/5
Write as a radical and then simplify.
21. 251/2
22. 361/2
23. 1211/2
24. 1441/2
25. (14)1/2
26. (49)1/2
27. 4−1/2
28. 9−1/2
29. (14)−1/2
30. (116)−1/2
31. 81/3
32. 1251/3
33. (127)1/3
34. (8125)1/3
35. (−27)1/3
36. (−64)1/3
37. 161/4
38. 6251/4
39. 81−1/4
40. 16−1/4
41. 100,0001/5
42. (−32)1/5
43. (132)1/5
44. (1243)1/5
45. 93/2
46. 43/2
47. 85/3
48. 272/3
49. 163/2
50. 322/5
51. (116)3/4
52. (181)3/4
53. (−27)2/3
54. (−27)4/3
55. (−32)3/5
56. (−32)4/5
Use a calculator to approximate an answer rounded to the nearest hundredth.
57. 23/4
58. 32/3
59. 51/5
60. 71/7
61. (−9)3/2
62. −93/2
63. Explain why (−4)^(3/2) gives an error on a calculator and −4^(3/2) gives an answer of −8.
64. Marcy received a text message from Mark asking her how old she was. In response, Marcy texted back “125^(2/3) years old.” Help Mark determine how old Marcy is.
Part B: Rational Exponents
Perform the operations and simplify. Leave answers in exponential form.
65. 22/3⋅24/3
66. 33/2⋅31/2
67. 51/2⋅51/3
68. 21/6⋅23/4
69. y1/4⋅y2/5
70. x1/2⋅x1/4
71. 57/351/3
72. 29/221/2
73. 2a2/3a1/6
74. 3b1/2b1/3
75. (81/2)2/3
76. (36)2/3
77. (x2/3)1/2
78. (y3/4)4/5
79. (4x2y4)1/2
80. (9x6y2)1/2
81. (2x1/3y2/3)3
82. (8x3/2y1/2)2
83. ( a 3/4 a 1/2)4/3
84. ( b 4/5 b 1/10)10/3
85. (4 x 2/3 y 4)1/2
86. (27 x 3/4 y 9)1/3
87. y1/2⋅y2/3y1/6
88. x2/5⋅x1/2x1/10
89. xyx1/2y1/3
90. x5/4yxy2/5
91. 49a5/7b3/27a3/7b1/4
92. 16a5/6b5/48a1/2b2/3
93. (9 x 2/3 y 6)3/2x1/2y
94. (125 x 3 y 3/5)2/3xy1/3
95. (27 a 1/4 b 3/2)2/3a1/6b1/2
96. (25 a 2/3 b 4/3)3/2a1/6b1/3
Part C: Mixed Indices
Perform the operations.
97. 93⋅35
98. 5⋅255
99. x⋅x3
100. y⋅y4
101. x23⋅x4
102. x35⋅x3
103. 100310
104. 16543
105. a23a
106. b45b3
107. x23x35
108. x34x23
109. 165
110. 93
111. 253
112. 553
113. 73
114. 33
Part D: Discussion Board
115. Who is credited for devising the notation for rational exponents? What are some of his other accomplishments?
116. When using text, it is best to communicate nth roots using rational exponents. Give an example.
Answers
1: 61/2
3: 111/3
5: 52/3
7: x1/5
9: x7/6
11: 2
13: 723
15: x34
17: 1x
19: x3
21: 5
23: 11
25: 1/2
27: 1/2
29: 2
31: 2
33: 1/3
35: −3
37: 2
39: 1/3
41: 10
43: 1/2
45: 27
47: 32
49: 64
51: 1/8
53: 9
55: −8
57: 1.68
59: 1.38
61: Not a real number
63: In the first expression, the square root of a negative number creates an error condition on the calculator. The square root of a negative number is not real. In the second expression, because of the order of operations, the negative sign is applied to the answer after 4 is raised to the (3/2) power.
65: 4
67: 55/6
69: y13/20
71: 25
73: 2a1/2
75: 2
77: x1/3
79: 2xy2
81: 8xy2
83: a1/3
85: 2x1/3y2
87: y
89: x1/2y2/3
91: 7x2/7y5/4
93: 27x1/2y8
95: 9b1/2
97: 31315
99: x56
101: x1112
103: 106
105: a6
107: x15
109: 45
111: 215
113: 76
8.6 Solving Radical Equations
Learning Objectives
1.Solve equations involving square roots.
2.Solve equations involving cube roots.
Radical Equations
A radical equationAny equation that contains one or more radicals with a variable in the radicand. is any equation that contains one or more radicals with a variable in the radicand. Following are some examples of radical equations, all of which will be solved in this section:
We begin with the squaring property of equalityGiven real numbers a and b, where a=b, then a2=b2.; given real numbers a and b, we have the following:
In other words, equality is retained if we square both sides of an equation.
The converse, on the other hand, is not necessarily true:
This is important because we will use this property to solve radical equations. Consider a very simple radical equation that can be solved by inspection:
Here we can see that x=9 is a solution. To solve this equation algebraically, make use of the squaring property of equality and the fact that (a)2=a2=a when a is positive. Eliminate the square root by squaring both sides of the equation as follows:
As a check, we can see that 9=3 as expected. Because the converse of the squaring property of equality is not necessarily true, solutions to the squared equation may not be solutions to the original. Hence squaring both sides of an equation introduces the possibility of extraneous solutionsA solution that does not solve the original equation., or solutions that do not solve the original equation. For this reason, we must check the answers that result from squaring both sides of an equation.
Example 1: Solve: x−1=5.
Solution: We can eliminate the square root by applying the squaring property of equality.
Next, we must check.
Answer: The solution is 26.
Example 2: Solve: 5−4x=x.
Solution: Begin by squaring both sides of the equation.
You are left with a quadratic equation that can be solved by factoring.
Since you squared both sides, you must check your solutions.
After checking, you can see that x=−5 was extraneous; it did not solve the original radical equation. Disregard that answer. This leaves x=1 as the only solution.
Answer: The solution is x=1.
In the previous two examples, notice that the radical is isolated on one side of the equation. Typically, this is not the case. The steps for solving radical equations involving square roots are outlined in the following example.
Example 3: Solve: 2x−5+4=x.
Solution:
Step 1: Isolate the square root. Begin by subtracting 4 from both sides of the equation.
Step 2: Square both sides. Squaring both sides eliminates the square root.
Step 3: Solve the resulting equation. Here you are left with a quadratic equation that can be solved by factoring.
Step 4: Check the solutions in the original equation. Squaring both sides introduces the possibility of extraneous solutions; hence the check is required.
After checking, we can see that x=3 is an extraneous root; it does not solve the original radical equation. This leaves x=7 as the only solution.
Answer: The solution is x=7.
Example 4: Solve: 3x+1−2x=0.
Solution: Begin by isolating the term with the radical.
Despite the fact that the term on the left side has a coefficient, it is still considered isolated. Recall that terms are separated by addition or subtraction operators.
Solve the resulting quadratic equation.
Since we squared both sides, we must check our solutions.
After checking, we can see that x=−34 was extraneous.
Answer: The solution is 3.
Sometimes both of the possible solutions are extraneous.
Example 5: Solve: 4−11x−x+2=0.
Solution: Begin by isolating the radical.
Since we squared both sides, we must check our solutions.
Since both possible solutions are extraneous, the equation has no solution.
Answer: No solution, Ø
The squaring property of equality extends to any positive integer power n. Given real numbers a and b, we have the following:
This is often referred to as the power property of equalityGiven any positive integer n and real numbers a and b, where a=b, then an=bn.. Use this property, along with the fact that (an)n=ann=a, when a is positive, to solve radical equations with indices greater than 2.
Example 6: Solve: x2+43−2=0.
Solution: Isolate the radical and then cube both sides of the equation.
Check.
Answer: The solutions are −2 and 2.
Try this! Solve: 2x−1+2=x.
Answer: x=5 (x=1 is extraneous)
Video Solution
(click to see video)
It may be the case that the equation has two radical expressions.
Example 7: Solve: 3x−4=2x+9.
Solution: Both radicals are considered isolated on separate sides of the equation.
Check x=13.
Answer: The solution is 13.
Example 8: Solve: x2+x−143=x+503.
Solution: Eliminate the radicals by cubing both sides.
Check.
Answer: The solutions are −8 and 8.
We will learn how to solve some of the more advanced radical equations in the next course, Intermediate Algebra.
Try this! Solve: 3x+1=2x−3.
Answer: 13
Video Solution
(click to see video)
Key Takeaways
•Solve equations involving square roots by first isolating the radical and then squaring both sides. Squaring a square root eliminates the radical, leaving us with an equation that can be solved using the techniques learned earlier in our study of algebra. However, squaring both sides of an equation introduces the possibility of extraneous solutions, so check your answers in the original equation.
•Solve equations involving cube roots by first isolating the radical and then cubing both sides. This eliminates the radical and results in an equation that may be solved with techniques you have already mastered.
Topic Exercises
Part A: Solving Radical Equations
Solve.
1. x=2
2. x=7
3. x+7=8
4. x+4=9
5. x+6=3
6. x+2=1
7. 5x−1=0
8. 3x−2=0
9. x−3=3
10. x+5=6
11. 3x+1=2
12. 5x−4=4
13. 7x+4+6=11
14. 3x−5+9=14
15. 2x−1−3=0
16. 3x+1−2=0
17. x3=2
18. x3=5
19. 2x+93=3
20. 4x−113=1
21. 5x+73+3=1
22. 3x−63+5=2
23. 2 x+23−1=0
24. 2 2x−33−1=0
25. 8x+11=3x+1
26. 23x−4=2(3x+1)
27. 2(x+10)=7x−15
28. 5(x−4)=x+4
29. 5x−23=4x3
30. 9(x−1)3=3(x+7)3
31. 3x+13=2(x−1)3
32. 9x3=3(x−6)3
33. 4x+21=x
34. 8x+9=x
35. 4(2x−3)=x
36. 3(4x−9)=x
37. 2x−1=x
38. 32x−9=x
39. 9x+9=x+1
40. 3x+10=x+4
41. x−1=x−3
42. 2x−5=x−4
43. 16−3x=x−6
44. 7−3x=x−3
45. 32x+10=x+9
46. 22x+5=x+4
47. 3x−1−1=x
48. 22x+2−1=x
49. 10x+41−5=x
50. 6(x+3)−3=x
51. 8x2−4x+1=2x
52. 18x2−6x+1=3x
53. 5x+2=x+8
54. 42(x+1)=x+7
55. x2−25=x
56. x2+9=x
57. 3+6x−11=x
58. 2+9x−8=x
59. 4x+25−x=7
60. 8x+73−x=10
61. 24x+3−3=2x
62. 26x+3−3=3x
63. 2x−4=14−10x
64. 3x−6=33−24x
65. x2−243=1
66. x2−543=3
67. x2+6x3+1=4
68. x2+2x3+5=7
69. 25x2−10x−73=−2
70. 9x2−12x−233=−3
71. 2x2−15x+25=(x+5)(x−5)
72. x2−4x+4=x(5−x)
73. 2(x2+3x−20)3=(x+3)23
74. 3x2+3x+403=(x−5)23
75. x1/2−10=0
76. x1/2−6=0
77. x1/3+2=0
78. x1/3+4=0
79. (x−1)1/2−3=0
80. (x+2)1/2−6=0
81. (2x−1)1/3+3=0
82. (3x−1)1/3−2=0
83. (4x+15)1/2−2x=0
84. (3x+2)1/2−3x=0
85. (2x+12)1/2−x=6
86. (4x+36)1/2−x=9
87. 2(5x+26)1/2=x+10
88. 3(x−1)1/2=x+1
89. The square root of 1 less than twice a number is equal to 2 less than the number. Find the number.
90. The square root of 4 less than twice a number is equal to 6 less than the number. Find the number.
91. The square root of twice a number is equal to one-
92. The square root of twice a number is equal to one-
93. The distance, d, measured in miles, a person can see an object is given by the formula d=6h2
where h represents the person’s height above sea level, measured in feet. How high must a person be to see an object 5 miles away?
94. The current, I, measured in amperes, is given by the formula I=PR
where P is the power usage, measured in watts, and R is the resistance, measured in ohms. If a light bulb requires 1/2 ampere of current and uses 60 watts of power, then what is the resistance of the bulb?
The period, T, of a pendulum in seconds is given by the formula T=2πL32
where L represents the length in feet. For each problem below, calculate the length of a pendulum, given the period. Give the exact value and the approximate value rounded off to the nearest tenth of a foot.
95. 1 second
96. 2 seconds
97. 1/2 second
98. 1/3 second
The time, t, in seconds an object is in free fall is given by the formula t=s4
where s represents the distance in feet the object has fallen. For each problem below, calculate the distance an object falls, given the amount of time.
99. 1 second
100. 2 seconds
101. 1/2 second
102. 1/4 second
The x-
103. y=x−3−1
104. y=x+2−3
105. y=x−13+2
106. y=x+13−3
Part B: Discussion Board
107. Discuss reasons why we sometimes obtain extraneous solutions when solving radical equations. Are there ever any conditions where we do not need to check for extraneous solutions? Why?
Answers
1: 4
3: 1
5: Ø
7: 1/25
9: 12
11: 1
13: 3
15: 13/4
17: 8
19: 9
21: −3
23: −15/8
25: 2
27: 7
29: 2
31: −3
33: 7
35: 2, 6
37: 2
39: −1, 8
41: 5
43: Ø
45: −3, 3
47: 2, 5
49: −4, −4
51: 1/2
53: 2, 7
55: Ø
57: 10
59: −6, −4
61: −1/2, 3/2
63: Ø
65: −5, 5
67: −9, 3
69: 1/5
71: 5, 10
73: −7, 7
75: 100
77: −8
79: 10
81: −13
83: 5/2
85: −6, −4
87: −2, 2
89: 5
91: 8
93: 1623 feet
95: 8/π2≈0.8 foot
97: 2/π2≈0.2 foot
99: 16 feet
101: 4 feet
103: (4, 0)
105: (−7, 0)
8.7 Review Exercises and Sample Exam
Review Exercises
(Assume all variables represent nonnegative numbers.)
Radicals
Simplify.
1. 36
2. 425
3. −16
4. −9
5. 1253
6. 3 −83
7. 1643
8. −5 −273
9. 40
10. −350
11. 9881
12. 1121
13. 5 1923
14. 2 −543
Simplifying Radical Expressions
Simplify.
15. 49x2
16. 25a2b2
17. 75x3y2
18. 200m4n3
19. 18x325y2
20. 108x349y4
21. 216x33
22. −125x6y33
23. 27a7b5c33
24. 120x9y43
Use the distance formula to calculate the distance between the given two points.
25. (5, −8) and (2, −10)
26. (−7, −1) and (−6, 1)
27. (−10, −1) and (0, −5)
28. (5, −1) and (−2, −2)
Adding and Subtracting Radical Expressions
Simplify.
29. 83+33
30. 1210−210
31. 143+52−53−62
32. 22ab−5ab+7ab−2ab
33. 7x−(3x+2y)
34. (8yx−7xy)−(5xy−12yx)
35. 45+12−20−75
36. 24−32+54−232
37. 23x2+45x−x27+20x
38. 56a2b+8a2b2−224a2b−a18b2
39. 5y4x2y−(x16y3−29x2y3)
40. (2b9a2c−3a16b2c)−(64a2b2c−9ba2c)
41. 216x3−125xy3−8x3
42. 128x33−2x⋅543+3 2x33
43. 8x3y3−2x⋅8y3+27x3y3+x⋅y3
44. 27a3b3−3 8ab33+a⋅64b3−b⋅a3
Multiplying and Dividing Radical Expressions
Multiply.
45. 3⋅6
46. (35)2
47. 2(3−6)
48. (2−6)2
49. (1−5)(1+5)
50. (23+5)(32−25)
51. 2a23⋅4a3
52. 25a2b3⋅5a2b23
Divide.
53. 724
54. 104864
55. 98x4y236x2
56. 81x6y738y33
Rationalize the denominator.
57. 27
58. 63
59. 142x
60. 1215
61. 12x23
62. 5a2b5ab23
63. 13−2
64. 2−62+6
Rational Exponents
Express in radical form.
65. 71/2
66. 32/3
67. x4/5
68. y−3/4
Write as a radical and then simplify.
69. 41/2
70. 501/2
71. 42/3
72. 811/3
73. (14)3/2
74. (1216)−1/3
Perform the operations and simplify. Leave answers in exponential form.
75. 31/2⋅33/2
76. 21/2⋅21/3
77. 43/241/2
78. 93/491/4
79. (36x4y2)1/2
80. (8x6y9)1/3
81. ( a 4/3 a 1/2)2/5
82. (16 x 4/3 y 2)1/2
Solving Radical Equations
Solve.
83. x=5
84. 2x−1=3
85. x−8+2=5
86. 3x−5−1=11
87. 5x−3=2x+15
88. 8x−15=x
89. x+41=x−1
90. 7−3x=x−3
91. 2(x+1)=2(x+1)
92. x(x+6)=4
93. x(3x+10)3=2
94. 2x2−x3+4=5
95. 3(x+4)(x+1)3=5x+373
96. 3x2−9x+243=(x+2)23
97. y1/2−3=0
98. y1/3+3=0
99. (x−5)1/2−2=0
100. (2x−1)1/3−5=0
Sample Exam
In problems 1–18, assume all variables represent nonnegative numbers.
1. Simplify.
1.100
2.−100
3.−100
2. Simplify.
1.273
2.−273
3.−273
3. 12825
4. 1921253
5. 512x2y3z
6. 250x2y3z53
Perform the operations.
7. 524−108+96−327
8. 38x2y−(x200y−18x2y)
9. 2ab(32a−b)
10. (x−2y)2
Rationalize the denominator.
11. 102x
12. 14xy23
13. 1x+5
14. 2−32+3
Perform the operations and simplify. Leave answers in exponential form.
15. 22/3⋅21/6
16. 104/5101/3
17. (121a4b2)1/2
18. (9 y 1/3 x 6)1/2y1/6
Solve.
19. x−7=0
20. 3x+5=1
21. 2x−1+2=x
22. 31−10x=x−4
23. (2x+1)(3x+2)=3(2x+1)
24. x(2x−15)3=3
25. The period, T, of a pendulum in seconds is given the formula T=2πL32, where L represents the length in feet. Calculate the length of a pendulum if the period is 1½ seconds. Round off to the nearest tenth.
Review Exercises Answers
1: 6
3: Not a real number
5: 5
7: 1/4
9: 210
11: 729
13: 20 33
15: 7x
17: 5xy3x
19: 3x2x5y
21: 6x
23: 3a2bc⋅ab23
25: 13
27: 229
29: 113
31: 93−2
33: 4x−2y
35: 5−33
37: −x3+55x
39: 12xyy
41: 4 x3−5 xy3
43: 2x⋅y3
45: 32
47: 6−23
49: −4
51: 2a
53: 32
55: 7xy26
57: 277
59: 72xx
61: 4x32x
63: 3+2
65: 7
67: x45
69: 2
71: 2 23
73: 1/8
75: 9
77: 4
79: 6x2y
81: a1/3
83: 25
85: 17
87: 6
89: 8
91: −1/2, −1
93: 2/3, −4
95: −5, 5/3
97: 9
99: 9
Sample Exam Answers
1:
1.10
2.Not a real number
3.−10
3: 825
5: 10xy3yz
7: 146−153
9: 6a2b−2ba
11: 52xx
13: x−5x−25
15: 25/6
17: 11a2b
19: 49
21: 5
23: −1/2, 1/3
25: 1.8 feet
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