1. No category

9.3
Adding and Subtracting Radicals
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.
NOTE “Indices” is the plural
of “index.”
3
12 and 12 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
Example 1
Adding and Subtracting Like Radicals
Simplify each expression.
NOTE Apply the distributive
(a) 512 3 12 (5 3)12 812
property, then combine the
coefficients.
(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
718
CHAPTER 9
EXPONENTS AND RADICALS
so
18
312 18 312 212
(3 2)12 512
NOTE Simplify 112.
(b) 513 112 513 14 3
513 14 13
NOTE The radicals can now be
513 213
combined. Do you see why?
(5 2)13 313
CHECK YOURSELF 2
Simplify.
(a) 12 118
(b) 5 13 127
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
(a) 513x 213x (5 2)13x 313x
involved, we apply the
distributive property and
combine terms as before.
(b) 223a3 5a13a
22a2 3a 5a13a
NOTE Simplify the first term.
22a 2 13a 5a13a
2a13a 5a13a
NOTE The radicals can now be
(2a 5a)13a 7a13a
combined.
Simplify each expression.
(a) 217y 317y
(b) 220a2 a145
CHECK YOURSELF ANSWERS
1. (a) 915; (b) 817; (c) 413
3. (a) 517y; (b) a15
2. (a) 412; (b) 213
© 2001 McGraw-Hill Companies
CHECK YOURSELF 3
Name
9.3
Exercises
Section
Date
Simplify by combining like terms.
1. 212 4 12
2. 13 513
ANSWERS
1.
2.
3. 1117 417
4. 5 13 312
3.
4.
5. 517 3 16
6. 3 15 515
5.
6.
7. 213 5 13
8. 2 111 5111
7.
8.
9. 213x 513x
10. 712a 312a
9.
10.
11.
11. 213 13 313
12. 315 215 15
12.
13.
13. 517 2 17 17
14. 3110 2110 110
14.
15.
15. 215x 515x 215x
16. 513b 213b 413b
16.
17.
17. 213 112
18. 512 118
18.
19.
© 2001 McGraw-Hill Companies
20.
19. 120 15
20. 198 312
21.
22.
21. 216 154
22. 213 127
23.
24.
23. 172 150
24. 127 112
719
ANSWERS
25.
25. 3112 148
26. 518 2118
27. 2145 2120
28. 2198 4118
29. 112 127 13
30. 150 132 18
31. 3124 154 16
32. 163 2128 517
33. 2150 3118 132
34. 3127 4112 1300
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
Simplify by combining like terms.
38.
2
35. a127 223a
2
36. 522y 3y18
37. 523x3 2127x
38. 722a3 18a
39.
40.
41.
42.
Use a calculator to find a decimal approximation for each of the following. Round your
answer to the nearest hundredth.
43.
44.
45.
39. 13 12
40. 17 111
41. 15 13
42. 117 113
43. 413 715
44. 812 317
45. 517 8113
46. 712 4111
720
© 2001 McGraw-Hill Companies
46.
ANSWERS
47. Perimeter of a rectangle. Find the perimeter of the rectangle shown in the figure.
47.
48.
36
49.
49
50.
a.
b.
48. Perimeter of a rectangle. Find the perimeter of the rectangle shown in the figure.
c.
Write your answer in radical form.
d.
147
e.
108
f.
g.
h.
49. Perimeter of a triangle. Find the perimeter of the triangle shown in the figure.
3
3 2
3 2
50. Perimeter of a triangle. Find the perimeter of the triangle shown in the figure.
4
© 2001 McGraw-Hill Companies
5 3
5 3
Getting Ready for Section 9.4 [Section 3.4]
Perform the indicated multiplication.
(a)
(c)
(e)
(g)
2(x 5)
m(m 8)
(w 2)(w 2)
(x y)(x y)
(b)
(d)
(f)
(h)
3(a 3)
y( y 7)
(x 3)(x 3)
(b 7)(b 7)
721
Answers
© 2001 McGraw-Hill Companies
1. 612
3. 717
5. Cannot be simplified
7. 313
9. 713x
11. 613
13. 417
15. 515x
17. 413
19. 15
21. 16
23. 1112
25. 213
27. 215
29. 413
31. 416
33. 1512
35. a13
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
b. 3a 9
2
c. m2 8m
d. y2 7y
e. w 2 4
f. x 2 9
g. x2 2xy y
2
h. b 14b 49
722