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This is “Adding and Subtracting Radical Expressions”, section 8.3 from the book Beginning Algebra (v. 1.0). For details on it (including licensing), click here.
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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
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