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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 n2, 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 radicalsabn=anbn, where a and b represent positive real numbers.:

abn=anbn

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: 363

 

Try this! Simplify: 2963.

Answer: 4123

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: 235

 

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: 343

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. 5273

54. 283

55. 51,0003

56. 32435

57. 10164

58. 2646

59. 325

60. 64

61. 2273

62. 82435

63. 783

64. 46254

65. 6100,0005

66. 51287

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. 5483

94. 21083

95. 8964

96. 71624

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. 273

118. 363

119. 254

120. 324

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: 323

85: 263

87: 253

89: 363

91: 3235

93: 1063

95: 1664

97: 255

99: 2753

101: 222.83

103: 356.71

105: 320.87

107: 4251.13

109: 21034.31

111: 2633.63

113: 60

115: 250

117: 563

119: 804

121: 32 units

123: 2143 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 =32x2=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    a0         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: 2xy210x2y3

 

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: x293yz3

 

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=b4b. Hence the factor b will be left inside the radical.

Answer: 3abb4

 

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=y5y; the factor y will be left inside the radical as well.

Answer: 2yzx3y5

 

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=a2a,     which is   a5÷2=a2r1b53=bb23,      which is    b5÷3=b1r2c145=c2c45,    which is   c14÷5=c2r4

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-values, and plot the resulting ordered pairs.

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-values.

Plot the points and sketch the graph of the cube root function.

 

Example 12: Given the function g(x)=x13, 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)=323

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(3x1)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. 2x8x63

54. 5x227x33

55. 2ab8a4b53

56. 5a2b27a3b33

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. 52x3

66. 35y3

67. 2x33

68. 3y23

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-intercepts for any graph will have the form (0, y), where y is a real number. Therefore, to find y-intercepts, set x = 0 and solve for y. Find the y-intercepts for the following.

83. y=x+41

84. y=x+13

85. y=x13+2

86. y=x+133

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. x26x+9

94. x210x+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)=x1, 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)=x5, 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)=x32, find g(1), g(0), and g(8)

114. g(x)=x13+2, find g(0), g(2), and g(9)

For each function, fill in the table.

115. f(x)=x+1

116. f(x)=x2

117. f(x)=x3+1

118. f(x)=x+23

Part D: Discussion Board

119. Give a value for x such that x2x. 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(3x1)

17: 3xx5y

19: m3m6n2

21: rs22s5t2

23: 3a

25: 5xy2x3

27: 4xy2z3

29: 2xyy3

31: abc2ab23

33: 2xx33y

35: 2rs4t445r2t3

37: 3x

39: 2xy2

41: abcbc24

43: 2x8x24

45: 2m2n

47: 6x

49: 10x2y

51: 12a3b2ab

53: 4x3

55: 4a2b2ab23

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: x3

95: 2x+3

97: 30 miles per hour

99: π3.1 seconds

101: π/40.8 seconds

103: 31.7 seconds

105: 233.5 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: 2737.

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+629572.

Solution:

We cannot simplify any further because 5 and 2 are not like radicals; the radicands are not the same.

Answer: 52

Caution

It is important to point out that 5252. We can verify this by calculating the value of each side with a calculator.

In general, note that an±bna±bn.

Example 4: Simplify: 363+266336.

Solution:

We cannot simplify any further because 63 and 6 are not like radicals; the indices are not the same.

Answer: 2636

 

Often we will have to simplify before we can identify the like radicals within the terms.

 

Example 5: Subtract: 1248.

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+2735212.

Solution:

Answer: 53

 

Try this! Subtract: 25068.

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: 62x33x3+72x3.

Solution:

We cannot combine any further because the remaining radical expressions do not share the same radicand; they are not like radicals. Note that 2x33x32x3x3.

Answer: 2x33x3

 

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: (9x2y)(10x+7y).

Solution:

Answer: x9y

 

Until we simplify, it is often unclear which terms involving radicals are similar.

 

Example 9: Simplify: 52y3(54y3163).

Solution:

Answer: 22y3+223 

 

Example 10: Simplify: 2a125a2ba280b+420a4b.

Solution:

Answer: 14a25b

 

Try this! Simplify: 45x3(20x380x).

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+y2x+yx3+y33x+y

The property abn=anbn 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. 4575

4. 310810

5. 646+26

6. 5101510210

7. 1376257+52

8. 10131215+5131815

9. 65(4335)

10. 122(66+2)

11. (25310)(10+35)

12. (83+615)(315)

13. 463353+663

14. 103+51034103

15. (793433)(93333)

16. (853+253)(253+6253)

Simplify. (Assume all radicands containing variable expressions are positive.)

17. 9x+7x

18. 8y+4y

19. 7xy3xy+xy

20. 10y2x12y2x2y2x

21. 2ab5a+6ab10a

22. 3xy+6y4xy7y

23. 5xy(3xy7xy)

24. 8ab(2ab4ab)

25. (32x3x)(2x73x)

26. (y42y)(y52y)

27. 5x312x3

28. 2y33y3

29. a3b5+4a3b5a3b5

30. 8ab4+3ab42ab4

31. 62a42a3+72a2a3

32. 43a5+3a393a5+3a3

33. (4xy4xy3)(24xy4xy3)

34. (56y65y)(26y6+3y)

Part B: Adding and Subtracting Rational Expressions

Simplify.

35. 7512

36. 2454

37. 32+278

38. 20+4845

39. 2827+6312

40. 90+244054

41. 4580+2455

42. 108+48753

43. 42(2772)

44. 35(2050)

45. 163543

46. 813243

47. 1353+40353

48. 108332343

49. 227212

50. 350432

51. 324321848

52. 6216224296

53. 218375298+448

54. 24512+220108

55. (2363396)(712254)

56. (2288+3360)(272740)

57. 3543+525034163

58. 416232384337503

Simplify. (Assume all radicands containing variable expressions are positive.)

59. 81b+4b

60. 100a+a

61. 9a2b36a2b

62. 50a218a2

63. 49x9y+x4y

64. 9x+64y25xy

65. 78x(316y218x)

66. 264y(332y81y)

67. 29m2n5m9n+m2n

68. 418n2m2n8m+n2m

69. 4x2y9xy216x2y+y2x

70. 32x2y2+12x2y18x2y227x2y

71. (9x2y16y)(49x2y4y)

72. (72x2y218x2y)(50x2y2+x2y)

73. 12m4nm75m2n+227m4n

74. 5n27mn2+212mn4n3mn2

75. 227a3ba48aba144a3b

76. 298a4b2a162a2b+a200b

77. 125a327a3

78. 1000a2364a23

79. 2x54x3216x43+52x43

80. x54x33250x63+x223

81. 16y24+81y24

82. 32y45y45

83. 32a34162a34+52a34

84. 80a4b4+5a4b4a5b4

85. 27x33+8x3125x33

86. 24x3128x381x3

87. 27x4y38xy33+x64xy3yx3

88. 125xy33+8x3y3216xy33+10xy3

89. (162x4y3250x4y23)(2x4y23384x4y3)

90. (32x2y65243x6y25)(x2y65xxy25)

Part C: Discussion Board

91. Choose values for x and y and use a calculator to show that x+yx+y.

92. Choose values for x and y and use a calculator to show that x2+y2x+y.

Answers

1: 143

3: 25

5: 6

7: 872

9: 9543

11: 5410

13: 1063353

15: 69333

17: 16x

19: 5xy

21: 8ab15a

23: 9xy

25: 22x+63x

27: 7x3

29: 4a3b5

31: 132a52a3

33: 4xy4

35: 33

37: 22+33

39: 5753

41: 55

43: 10233

45: 23

47: 453

49: 23

51: 23362

53: 82+3

55: 8366

57: 2623

59: 11b

61: 3ab

63: 8x5y

65: 202x12y

67: 8mn

69: 2xy2yx

71: 4xy

73: 3m23n

75: 2a3ab12a2ab

77: 2a3

79: 7x2x3

81: 5y24

83: 42a34

85: 2x+2x3

87: 7xxy33yx3

89: 7x6xy36x2xy23

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: 26.

Solution: This problem is a product of two square roots. Apply the product rule for radicals and then simplify.

Answer: 23

 

Example 2: Multiply: 9363.

Solution: This problem is a product of cube roots. Apply the product rule for radicals and then simplify.

Answer: 323

 

Often there will be coefficients in front of the radicals.

 

Example 3: Multiply: 2352.

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: 25x3325x23.

Solution:

Answer: 30x

 

Use the distributive property when multiplying rational expressions with more than one term.

 

Example 5: Multiply: 43(2336).

Solution: Apply the distributive property and multiply each term by 43.

Answer: 24362

 

Example 6: Multiply: 4x23(2x354x23).

Solution: Apply the distributive property and then simplify the result.

Answer: 2x10x2x3

 

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)(54).

Solution: Begin by applying the distributive property.

Answer: 325

 

Example 8: Multiply: (3xy)2.

Solution:

Answer: 9x6xy+y

 

Try this! Multiply: (23+52)(326).

Answer: 6122+56203

Video Solution

(click to see video)

The expressions (a+b) and (ab) are called conjugatesThe factors (a+b) and (ab) are conjugates.. When multiplying conjugates, the sum of the products of the inner and outer terms results in 0.

 

Example 9: Multiply: (2+5)(25).

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 b0, 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: 3ba32

 

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: 14x35.

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: 132.

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)(ab)=ab to expedite the process of multiplying the expressions in the denominator.

 

Example 20: Rationalize the denominator: 262+6.

Solution: Multiply by 1 in the form 2626.

Answer: 2+3

 

Example 21: Rationalize the denominator: x+yxy.

Solution: In this example, we will multiply by 1 in the form xyxy.

Answer: x2xy+yxy

 

Try this! Rationalize the denominator: 35+5253.

Answer: 195+4511

Video Solution

(click to see video)

Key Takeaways

  • To multiply two single-term radical expressions, multiply the coefficients and multiply the radicands. If possible, simplify the result.
  • Apply the distributive property when multiplying radical expressions with multiple terms. Then simplify and combine all like radicals.
  • Multiplying a two-term radical expression involving square roots by its conjugate results in a rational expression.
  • 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. 35

2. 73

3. 26

4. 515

5. 77

6. 1212

7. 25710

8. 31526

9. (25)2

10. (62)2

11. 2x2x

12. 5y5y

13. 3a12

14. 3a2a

15. 42x36x

16. 510y22y

17. 53253

18. 4323

19. 43103

20. 18363

21. (593)(263)

22. (243)(343)

23. (223)3

24. (343)3

25. 3a239a3

26. 7b349b23

27. 6x234x23

28. 12y39y23

29. 20x2y310x2y23

30. 63xy312x4y23

31. 5(35)

32. 2(32)

33. 37(273)

34. 25(6310)

35. 6(32)

36. 15(5+3)

37. x(x+xy)

38. y(xy+y)

39. 2ab(14a210b)

40. 6ab(52a3b)

41. (25)(3+7)

42. (3+2)(57)

43. (234)(36+1)

44. (526)(723)

45. (53)2

46. (72)2

47. (23+2)(232)

48. (2+37)(237)

49. (a2b)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. 353

72. 622

73. 17x

74. 13y

75. a5ab

76. 3b223ab

77. 2363

78. 1473

79. 14x3

80. 13y23

81. 9x239xy23

82. 5y2x35x2y3

83. 3a23a2b23

84. 25n325m2n3

85. 327x2y5

86. 216xy25

87. ab9a3b5

88. abcab2c35

89. 3103

90. 262

91. 15+3

92. 172

93. 33+6

94. 55+15

95. 10535

96. 22432

97. 3+535

98. 10210+2

99. 233243+2

100. 65+2252

101. x+yxy

102. xyx+y

103. aba+b

104. ab+2ab2

105. x52x

106. 1xy

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: 253

21: 3023

23: 16

25: 3a

27: 2x3x3

29: 2xy25x3

31: 355

33: 42321

35: 3223

37: x+xy

39: 2a7b4b5a

41: 6+141535

43: 182+231264

45: 8215

47: 10

49: a22ab+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: 3153

73: 7x7x

75: ab5b

77: 633

79: 2x232x

81: 36x2y3y

83: 9ab32b

85: 9x3y45xy

87: 27a2b453

89: 310+9

91: 532

93: 1+2

95: 5352

97: 415

99: 157623

101: x2+2xy+yx2y

103: a2ab+bab

105: 5x+2x254x

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: xmxn=xm+n
Quotient rule: xmxn=xmn,x0
Power rule: (xm)n=xmn
Power rule for a product: (xy)n=xnyn
Power rule for a quotient: (xy)n=xnyn,y0
Negative exponents: xn=1xn
Zero exponent: x0=1,x0

These rules allow us to perform operations with rational exponents.

 

Example 9: Simplify: 22/321/6.

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: 253/2.

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: 223.

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. x1/2

18. x3/4

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. 41/2

28. 91/2

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. 811/4

40. 161/4

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. 22/324/3

66. 33/231/2

67. 51/251/3

68. 21/623/4

69. y1/4y2/5

70. x1/2x1/4

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. y1/2y2/3y1/6

88. x2/5x1/2x1/10

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. 9335

98. 5255

99. xx3

100. yy4

101. x23x4

102. x35x3

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: x1=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: 54x=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: 2x5+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+12x=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: 411xx+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+432=0.

Solution: Isolate the radical and then cube both sides of the equation.

Check.

Answer: The solutions are −2 and 2.

 

Try this! Solve: 2x1+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: 3x4=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+x143=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=2x3.

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. 5x1=0

8. 3x2=0

9. x3=3

10. x+5=6

11. 3x+1=2

12. 5x4=4

13. 7x+4+6=11

14. 3x5+9=14

15. 2x13=0

16. 3x+12=0

17. x3=2

18. x3=5

19. 2x+93=3

20. 4x113=1

21. 5x+73+3=1

22. 3x63+5=2

23. 2x+231=0

24. 22x331=0

25. 8x+11=3x+1

26. 23x4=2(3x+1)

27. 2(x+10)=7x15

28. 5(x4)=x+4

29. 5x23=4x3

30. 9(x1)3=3(x+7)3

31. 3x+13=2(x1)3

32. 9x3=3(x6)3

33. 4x+21=x

34. 8x+9=x

35. 4(2x3)=x

36. 3(4x9)=x

37. 2x1=x

38. 32x9=x

39. 9x+9=x+1

40. 3x+10=x+4

41. x1=x3

42. 2x5=x4

43. 163x=x6

44. 73x=x3

45. 32x+10=x+9

46. 22x+5=x+4

47. 3x11=x

48. 22x+21=x

49. 10x+415=x

50. 6(x+3)3=x

51. 8x24x+1=2x

52. 18x26x+1=3x

53. 5x+2=x+8

54. 42(x+1)=x+7

55. x225=x

56. x2+9=x

57. 3+6x11=x

58. 2+9x8=x

59. 4x+25x=7

60. 8x+73x=10

61. 24x+33=2x

62. 26x+33=3x

63. 2x4=1410x

64. 3x6=3324x

65. x2243=1

66. x2543=3

67. x2+6x3+1=4

68. x2+2x3+5=7

69. 25x210x73=2

70. 9x212x233=3

71. 2x215x+25=(x+5)(x5)

72. x24x+4=x(5x)

73. 2(x2+3x20)3=(x+3)23

74. 3x2+3x+403=(x5)23

75. x1/210=0

76. x1/26=0

77. x1/3+2=0

78. x1/3+4=0

79. (x1)1/23=0

80. (x+2)1/26=0

81. (2x1)1/3+3=0

82. (3x1)1/32=0

83. (4x+15)1/22x=0

84. (3x+2)1/23x=0

85. (2x+12)1/2x=6

86. (4x+36)1/2x=9

87. 2(5x+26)1/2=x+10

88. 3(x1)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-half of that number. Find the number.

92. The square root of twice a number is equal to one-third of that number. Find the number.

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-intercepts for any graph have the form (x, 0), where x is a real number. Therefore, to find x-intercepts, set y = 0 and solve for x. Find the x-intercepts for each of the following.

103. y=x31

104. y=x+23

105. y=x13+2

106. y=x+133

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/π20.8 foot

97: 2/π20.2 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. 383

7. 1643

8. 5273

9. 40

10. 350

11. 9881

12. 1121

13. 51923

14. 2543

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. 1210210

31. 143+525362

32. 22ab5ab+7ab2ab

33. 7x(3x+2y)

34. (8yx7xy)(5xy12yx)

35. 45+122075

36. 2432+54232

37. 23x2+45xx27+20x

38. 56a2b+8a2b2224a2ba18b2

39. 5y4x2y(x16y329x2y3)

40. (2b9a2c3a16b2c)(64a2b2c9ba2c)

41. 216x3125xy38x3

42. 128x332x543+32x33

43. 8x3y32x8y3+27x3y3+xy3

44. 27a3b338ab33+a64b3ba3

Multiplying and Dividing Radical Expressions

Multiply.

45. 36

46. (35)2

47. 2(36)

48. (26)2

49. (15)(1+5)

50. (23+5)(3225)

51. 2a234a3

52. 25a2b35a2b23

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. 132

64. 262+6

Rational Exponents

Express in radical form.

65. 71/2

66. 32/3

67. x4/5

68. y3/4

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. 31/233/2

76. 21/221/3

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. 2x1=3

85. x8+2=5

86. 3x51=11

87. 5x3=2x+15

88. 8x15=x

89. x+41=x1

90. 73x=x3

91. 2(x+1)=2(x+1)

92. x(x+6)=4

93. x(3x+10)3=2

94. 2x2x3+4=5

95. 3(x+4)(x+1)3=5x+373

96. 3x29x+243=(x+2)23

97. y1/23=0

98. y1/3+3=0

99. (x5)1/22=0

100. (2x1)1/35=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. 524108+96327

8. 38x2y(x200y18x2y)

9. 2ab(32ab)

10. (x2y)2

Rationalize the denominator.

11. 102x

12. 14xy23

13. 1x+5

14. 232+3

Perform the operations and simplify. Leave answers in exponential form.

15. 22/321/6

16. 104/5101/3

17. (121a4b2)1/2

18. (9 y 1/3 x 6)1/2y1/6

Solve.

19. x7=0

20. 3x+5=1

21. 2x1+2=x

22. 3110x=x4

23. (2x+1)(3x+2)=3(2x+1)

24. x(2x15)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: 2033

15: 7x

17: 5xy3x

19: 3x2x5y

21: 6x

23: 3a2bcab23

25: 13

27: 229

29: 113

31: 932

33: 4x2y

35: 533

37: x3+55x

39: 12xyy

41: 4x35xy3

43: 2xy3

45: 32

47: 623

49: −4

51: 2a

53: 32

55: 7xy26

57: 277

59: 72xx

61: 4x32x

63: 3+2

65: 7

67: x45

69: 2

71: 223

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: 146153

9: 6a2b2ba

11: 52xx

13: x5x25

15: 25/6

17: 11a2b

19: 49

21: 5

23: −1/2, 1/3

25: 1.8 feet