9.4
Multiplying and Dividing Radicals
9.4
OBJECTIVES
1. Multiply and divide expressions involving numeric
radicals
2. Multiply and divide expressions involving
algebraic radicals
In Section 9.2 we stated the first property for radicals:
1ab 1a 1b
when a and b are any positive real numbers
That property has been used to simplify radical expressions up to this point. Suppose now
that we want to find a product, such as 13 15.
We can use our first radical rule in the opposite manner.
NOTE The product of square
1a 1b 1ab
roots is equal to the square root
of the product of the radicands.
so
13 15 13 5 115
We may have to simplify after multiplying, as Example 1 illustrates.
Example 1
Simplifying Radical Expressions
Multiply then simplify each expression.
(a) 15 110 15 10 150
125 2 512
(b) 112 16 112 6 172
136 2 136 12 612
An alternative approach would be to simplify 112 first.
112 16 213 16 2118
219 2 2 19 12
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2 312 6 12
(c) 110x 12x 220x2 24x2 5
24x2 15 2x15
723
724
CHAPTER 9
EXPONENTS AND RADICALS
CHECK YOURSELF 1
Simplify.
(a) 13 16
(c) 18a 13a
(b) 13 118
If coefficients are involved in a product, we can use the commutative and associative
properties to change the order and grouping of the factors. This is illustrated in Example 2.
Example 2
Multiplying Radical Expressions
Multiply.
(215)(3 16) (2 3)(15 16)
necessary to show the
intermediate steps.
6 15 6
6130
CHECK YOURSELF 2
Multiply (317)(513).
The distributive property can also be applied in multiplying radical expressions.
Consider the following.
Example 3
Multiplying Radical Expressions
Multiply.
(a) 13(12 13)
13 12 13 13
The distributive property
16 3
Multiply the radicals.
(b) 15(216 313)
15 216 15 313
The distributive property
2 15 16 3 15 13
The commutative property
2130 3 115
CHECK YOURSELF 3
Multiply.
(a) 15(16 15)
(b) 13(215 312)
The FOIL pattern we used for multiplying binomials in Section 3.4 can also be applied
in multiplying radical expressions. This is shown in Example 4.
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NOTE In practice, it is not
MULTIPLYING AND DIVIDING RADICALS
SECTION 9.4
725
Example 4
Multiplying Radical Expressions
Multiply.
(a) (13 2)(13 5)
13 13 513 213 2 5
3 513 213 10
Combine like terms.
13 713
C A U TI O N
NOTE You can use the pattern
(a b)(a b) a2 b2, where
a 17 and b 2, for the same
result. 17 2 and 17 2 are
called conjugates of each other.
Note that their product is the
rational number 3. The product
of conjugates will always be
rational.
Be Careful! This result cannot be further simplified: 13 and 7 13 are not like terms.
(b) (17 2)(17 2) 17 17 2 17 217 4
743
(c) (13 5)2 (13 5)(13 5)
13 13 513 513 5 5
3 513 513 25
28 1013
CHECK YOURSELF 4
Multiply.
(a) (15 3)(15 2)
(b) (13 4)(13 4)
(c) (12 3)2
We can also use our second property for radicals in the opposite manner.
NOTE The quotient of square
roots is equal to the square root
of the quotient of the
radicands.
1a
a
1b
Ab
One use of this property to divide radical expressions is illustrated in Example 5.
Example 5
Simplifying Radical Expressions
Simplify.
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NOTE The clue to recognizing
(a)
148
48
116 4
13
A3
(b)
1200
200
1100 10
12
A 2
(c)
2125x2
125x2
225x2 5x
15
A 5
when to use this approach is in
noting that 48 is divisible by 3.
There is one final quotient form that you may encounter in simplifying expressions, and
it will be extremely important in our work with quadratic equations in the next chapter. This
form is shown in Example 6.
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CHAPTER 9
EXPONENTS AND RADICALS
CHECK YOURSELF 5
Simplify.
(a)
175
13
(b)
281s2
19
Example 6
Simplifying Radical Expressions
Simplify the expression
3 172
3
Be Careful! Students are
sometimes tempted to write
3 612
1 612
3
First, we must simplify the radical in the numerator.
3 172
3 136 2
3
3
This is not correct. We must
divide both terms of the
numerator by the common
factor.
3 136 12
3 612
3
3
3(1 212)
1 212
3
Use Property 1 to simplify 172.
Factor the numerator—then divide
by the common factor 3.
CHECK YOURSELF 6
Simplify
15 175
.
5
CHECK YOURSELF ANSWERS
1. (a) 312; (b) 316; (c) 2a16
2. 15121
3. (a) 130 5;
(b) 2115 3 16
4. (a) 1 15; (b) 13; (c) 11 612
5. (a) 5; (b) 3s
6. 3 13
© 2001 McGraw-Hill Companies
C A U TI ON
Name
9.4 Exercises
Section
Date
Perform the indicated multiplication. Then simplify each radical expression.
1. 17 15
2. 13 17
ANSWERS
1.
2.
3. 15 111
4. 113 15
3.
4.
5. 13 110m
6. 17a 113
5.
6.
7. 12x 115
8. 117 12b
7.
8.
9.
9. 13 1 7 12
10. 15 17 13
10.
11.
11. 13 112
12. 17 17
12.
13.
13. 110 110
14. 15 115
14.
15.
16.
15. 118 16
16. 18 110
17.
18.
17. 12x 16x
18. 13a 115a
19.
© 2001 McGraw-Hill Companies
20.
21.
19. 213 17
20. 312 15
22.
21. (313)(5 17)
22. (2 15)(3 111)
727
ANSWERS
23.
23. (315)(2 110)
24. (4 13)(3 16)
25. 15(12 15)
26. 13(15 13)
27. 13(215 313)
28. 17(213 317)
29. (13 5)(13 3)
30. (15 2)(15 1)
31. (15 1)(15 3)
32. (12 3)(12 7)
33. (15 2)(15 2)
34. (17 5)(17 5)
35. (110 5)(110 5)
36. (111 3)(111 3)
37. (1x 3)(1x 3)
38. (1a 4)(1a 4)
2
39. (13 2)
2
40. (15 3)
2
41. (1y 5)
2
42. (1x 4)
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
45.
Perform the indicated division. Rationalize the denominator if necessary. Then simplify
each radical expression.
46.
728
43.
198
12
44.
1108
13
45.
272a2
12
46.
248m2
13
© 2001 McGraw-Hill Companies
44.
ANSWERS
4 148
47.
4
49.
51.
12 1108
48.
6
5 1175
5
50.
8 1512
4
52.
47.
48.
18 1567
9
49.
9 1108
3
51.
50.
52.
53.
6 118
3
53.
6 120
2
53.
54.
15 175
55.
5
8 148
56.
4
55.
56.
57. Area of a rectangle. Find the area of the rectangle shown in the figure.
3
57.
58.
11
59.
58. Area of a rectangle. Find the area of the rectangle shown in the figure.
3 5
60.
3 5
61.
59. Complete this statement: “ 12 15 110 because . . . .”
60. Explain why 213 513 713 but 713 315 1018.
61. When you look out over an unobstructed landscape or seascape, the distance to the
© 2001 McGraw-Hill Companies
visible horizon depends on your height above the ground. The equation
d
3
h
A2
is a good estimate of this, in which d distance to horizon in miles and h height
of viewer above the ground. Work with a partner to make a chart of distances to the
horizon given different elevations. Use the actual heights of tall buildings or
729
ANSWERS
a.
prominent landmarks in your area. The local library should have a list of these. Be
sure to consider the view to the horizon you get when flying in a plane. What would
your elevation have to be to see from one side of your city or town to the other? From
one side of your state or county to the other?
b.
c.
d.
e.
Getting Ready for Section 9.5 [Section 9.1]
f.
Evaluate the following. Round your answer to the nearest thousandth.
(a) 116
(c) 1121
(e) 127
(b) 149
(d) 112
(f) 198
Answers
1. 135
3. 155
13. 10
15. 613
23. 30 12
25. 110
31. 2 215
33. 1
41. y 101y 25
51. 2 412
53.
59.
61.
5.196
f.
9. 142
11. 6
21. 15121
29. 18 813
39. 7 413
13
49. 1 17
57. 133
c. 11
d. 3.464
9.899
© 2001 McGraw-Hill Companies
e.
5. 130m
7. 130x
17. 2x13
19. 2121
5
27. 2115 9
35. 15
37. x 9
43. 7
45. 6a
47. 1
2 12
55. 3 13
a. 4
b. 7
730