9.3 Adding and Subtracting Radicals
[Pages:6]9.3 Adding and Subtracting Radicals
NOTE "Indices" is the plural of "index."
9.3 OBJECTIVES
1. Add and subtract expressions involving numeric radicals
2. Add and subtract expressions involving algebraic radicals
Two radicals that have the same index and the same radicand (the expression inside the radical) are called like radicals. For example,
213 and 513 are like radicals.
12 and 15 are not like radicals--they have different radicands. 12 and 13 2 are not like radicals--they have different indices (2 and 3, representing a square root and a cube root).
Like radicals can be added (or subtracted) in the same way as like terms. We apply the distributive property and then combine the coefficients:
215 315 (2 3)15 515
NOTE Apply the distributive property, then combine the coefficients.
Example 1 Adding and Subtracting Like Radicals Simplify each expression.
(a) 512 3 12 (5 3)12 812 (b) 715 215 (7 2)15 515 (c) 817 17 217 (8 1 2)17 917
CHECK YOURSELF 1 Simplify.
(a) 215 7 15 (c) 513 213 13
(b) 917 17
? 2001 McGraw-Hill Companies
If a sum or difference involves terms that are not like radicals, we may be able to combine terms after simplifying the radicals according to our earlier methods.
Example 2 Adding and Subtracting Radicals Simplify each expression. (a) 312 18 We do not have like radicals, but we can simplify 18. Remember that 18 14 2 212
717
? 2001 McGraw-Hill Companies
718
CHAPTER 9 EXPONENTS AND RADICALS
NOTE Simplify 112.
NOTE The radicals can now be combined. Do you see why?
so
18
312 18 312 212 (3 2)12 512
(b) 513 112 513 14 3 513 14 13 513 213 (5 2)13 313
CHECK YOURSELF 2 Simplify.
(a) 12 118
(b) 513 127
If variables are involved in radical expressions, the process of combining terms proceeds in a fashion similar to that shown in previous examples. Consider Example 3. We again assume that all variables represent positive real numbers.
Example 3 Simplifying Expressions Involving Variables Simplify each expression.
NOTE Because like radicals are involved, we apply the distributive property and combine terms as before.
NOTE Simplify the first term.
NOTE The radicals can now be combined.
(a) 513x 213x (5 2)13x 313x
(b) 2 23a3 5a 13a 2 2a2 3a 5a 13a 2 2a 2 13a 5a 13a 2a13a 5a13a (2a 5a)13a 7a13a
CHECK YOURSELF 3 Simplify each expression.
(a) 217y 317y
(b) 220a2 a 145
CHECK YOURSELF ANSWERS
1. (a) 915; (b) 817; (c) 413 2. (a) 412; (b) 213 3. (a) 517y; (b) a15
? 2001 McGraw-Hill Companies
9.3 Exercises
Simplify by combining like terms. 1. 2 12 4 12 3. 11 17 4 17 5. 5 17 3 16 7. 2 13 5 13 9. 2 13x 5 13x 11. 2 13 13 3 13 13. 5 17 2 17 17 15. 2 15x 5 15x 2 15x 17. 2 13 112 19. 120 15 21. 2 16 154 23. 172 150
2. 13 5 13 4. 5 13 3 12 6. 3 15 5 15 8. 2 111 5 111 10. 7 12a 3 12a 12. 3 15 2 15 15 14. 3 110 2 110 110 16. 5 13b 2 13b 4 13b 18. 5 12 118 20. 198 3 12 22. 2 13 127 24. 127 112
Name Section
Date
ANSWERS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
719
ANSWERS 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.
720
25. 3 112 148
26. 5 18 2 118
27. 2 145 2 120
28. 2 198 4 118
29. 112 127 13
30. 150 132 18
31. 3 124 154 16
32. 163 2 128 5 17
33. 2 150 3 118 132 Simplify by combining like terms. 35. a 127 2 23a2
34. 3 127 4 112 1300 36. 5 22y2 3y 18
37. 5 23x3 2 127x
38. 7 22a3 18a
Use a calculator to find a decimal approximation for each of the following. Round your answer to the nearest hundredth.
39. 13 12
40. 17 111
41. 15 13
42. 117 113
43. 4 13 7 15
44. 8 12 3 17
45. 5 17 8 113
46. 7 12 4 111
? 2001 McGraw-Hill Companies
47. Perimeter of a rectangle. Find the perimeter of the rectangle shown in the figure.
36 49
48. Perimeter of a rectangle. Find the perimeter of the rectangle shown in the figure. Write your answer in radical form.
147 108
49. Perimeter of a triangle. Find the perimeter of the triangle shown in the figure.
ANSWERS 47. 48. 49. 50. a. b. c. d. e. f. g. h.
3 3 2
3 2
50. Perimeter of a triangle. Find the perimeter of the triangle shown in the figure.
5 3
4 5 3
? 2001 McGraw-Hill Companies
Getting Ready for Section 9.4 [Section 3.4]
Perform the indicated multiplication.
(a) 2(x 5) (c) m(m 8) (e) (w 2)(w 2) (g) (x y)(x y)
(b) 3(a 3) (d) y( y 7) (f) (x 3)(x 3) (h) (b 7)(b 7)
721
Answers
1. 6 12 3. 7 17 5. Cannot be simplified 7. 3 13 9. 7 13x
11. 6 13
13. 4 17
15. 5 15x
17. 4 13
19. 15
21. 16
23. 11 12
25. 2 13
27. 2 15
29. 4 13
31. 4 16
33. 15 12
35. a 13 37. (5x 6) 13x 39. 0.32 41. 3.97 43. 8.72
45. 42.07
47. 26
49. 2 13 3
a. 2x 10
c. m2 8m
d. y2 7y
e. w 2 4
f. x2 9
b. 3a 9 g. x2 2xy y 2
h. b2 14b 49
? 2001 McGraw-Hill Companies
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