Section 9.4
Radical Expressions
941
9.4 Exercises
In Exercises 1-14, place each of the radical expressions in simple radical form.
Check your answer with your calculator.
√
1. 2(5 7)
√
2. −3(2 3)
√ √
3. − 3(2 5)
√ √
4.
2(3 7)
√ √
5.
3(5 6)
√
√
6.
2(−3 10)
√
√
7. (2 5)(−3 3)
√
√
8. (−5 2)(−2 7)
√
√
9. (−4 3)(2 6)
√
√
10. (2 5)(−3 10)
√
11. (2 3)2
√
12. (−3 5)2
√
13. (−5 2)2
√
14. (7 11)2
In Exercises 15-22, use the distributive
property to multiply. Place your final
answer in simple radical form. Check
your result with your calculator.
√
15. 2(3 + 5)
√
16. −3(4 − 7)
√
17. 2(−5 + 4 2)
1
18.
19.
20.
√
−3(4 − 3 2)
√
√
2(2 +
3(4 −
√
2)
√
21.
6)
√
√ √
2( 10 + 14)
22.
√ √
√
3( 15 − 33)
In Exercises 23-30, combine like terms.
Place your final answer in simple radical
form. Check your solution with your calculator.
23.
√
√
−5 2 + 7 2
24.
√
√
2 3+3 3
25.
26.
27.
28.
29.
30.
√
√
2 6−8 6
√
√
7−3 7
√
√
√
2 3−4 2+3 3
√
√
√
7 5+2 7−3 5
√
√
√
√
2 3+5 2−7 3+2 2
√
√
√
√
3 11 − 2 7 − 2 11 + 4 7
In Exercises 31-40, combine like terms
where possible. Place your final answer
in simple radical form. Use your calculator to check your result.
√
√
31.
45 + 20
√
√
32. −4 45 − 4 20
33.
√
√
2 18 − 8
Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/
Version: Fall 2007
942
34.
35.
36.
37.
38.
39.
40.
Chapter 9
Radical Functions
√
√
− 20 + 4 45
√
√
−5 27 + 5 12
√
√
3 12 − 2 27
√
√
4 20 + 4 45
√
√
−2 18 − 5 8
√
√
2 45 + 5 20
√
√
3 27 − 4 12
In Exercises 41-48, simplify each of the
given rational expressions. Place your final answer in simple radical form. Check
your result with your calculator.
41.
√
1
2− √
2
42.
√
3
3 3− √
3
43.
√
2
2 2− √
2
44.
√
5
4 5− √
5
45.
√
3
5 2+ √
2
46.
√
2
6 3+ √
3
47.
48.
√
√
√
12
8− √ −3 2
2
√
6
27 − √ − 5 3
3
Version: Fall 2007
In Exercises 49-60, multiply to expand
each of the given radical expressions. Place
your final answer in simple radical form.
Use your calculator to check your result.
√
3)(3 −
√
49.
(2 +
50.
(5 +
51.
√
√
(4 + 3 2)(2 − 5 2)
52.
√
√
(3 + 5 3)(1 − 2 3)
53.
√
√
(2 + 3 2)(2 − 3 2)
54.
√
√
(3 + 2 5)(3 − 2 5)
55.
√
√
√
√
(2 3 + 3 2)(2 3 − 3 2)
56.
√
√
√
√
(8 2 + 5)(8 2 − 5)
57.
(2 +
58.
(3 −
59.
√
√
( 3 − 2 5)2
60.
√
√
(2 3 + 3 2)2
√
√
√
2)(2 −
√
3)
2)
5)2
2)2
In Exercises 61-68, place each of the
given rational expressions in simple radical form by “rationalizing the denominator.” Check your result with your calculator.
1
√
5+ 3
61.
√
62.
1
√
2 3− 2
63.
6
√
2 5− 2
64.
9
√
3 3− 6
√
√
√
Section 9.4
65.
66.
67.
68.
√
2+ 3
√
2− 3
√
3− 5
√
3+ 5
√
√
3+ 2
√
√
3− 2
√
√
2 3+ 2
√
√
3− 2
Radical Expressions
78. Given f (x) =
expression
√
943
x + 2, evaluate the
f (x) − f (3)
,
x−3
and then “rationalize the numerator.”
79. Given f (x) =
pression
√
x, evaluate the ex-
f (x + h) − f (x)
,
h
In Exercises 69-76, use the quadratic
formula to find the solutions of the given
equation. Place your solutions in simple
radical form and reduce your solutions to
lowest terms.
69.
3x2 − 8x = 5
70.
5x2 − 2x = 1
71.
5x2 = 2x + 1
72.
3x2 − 2x = 11
73.
7x2 = 6x + 2
74.
11x2 + 6x = 4
75.
x2 = 2x + 19
76.
100x2 = 40x − 1
and then “rationalize the numerator.”
80. Given f (x) =
expression
√
x − 3, evaluate the
f (x + h) − f (x)
,
h
and then “rationalize the numerator.”
In Exercises 77-80, we will suspend the
usual rule that you should rationalize the
denominator. Instead, just this one time,
rationalize the numerator of the resulting
expression.
77. Given f (x) =
pression
√
x, evaluate the ex-
f (x) − f (2)
,
x−2
and then “rationalize the numerator.”
Version: Fall 2007
944
Chapter 9
Radical Functions
9.4 Answers
1.
√
10 7
45.
√
13 2/2
3.
√
−2 15
47.
√
−7 2
5.
√
15 2
49.
3+
7.
√
−6 15
51.
√
−22 − 14 2
9.
√
−24 2
53.
−14
√
3
11.
12
55.
−6
13.
50
57.
√
9+4 5
15.
√
6+2 5
59.
17.
√
−10 + 8 2
19.
√
2 2+2
21.
√
√
2 5+2 7
63.
23.
√
2 2
65.
√
23 − 4 15
√
√
5− 3
2
√
√
2 5+ 2
3
√
7+4 3
25.
√
−6 6
67.
√
5+2 6
27.
√
√
5 3−4 2
69.
(4 ±
29.
√
√
7 2−5 3
71.
(1 ±
31.
√
5 5
73.
(3 ±
33.
√
4 2
75.
√
1±2 5
35.
√
−5 3
77.
√
37.
√
20 5
1
√
x+ 2
79.
√
1
√
x+h+ x
39.
41.
43.
√
16 5
√
√
2/2
2
Version: Fall 2007
61.
√
√
√
31)/3
6)/5
23)/7