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

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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

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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

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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.

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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

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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

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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)

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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

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