CHAPTER 1 Introductory Information and Review
Section 1.4:
Exponents and Radicals
 Evaluating Exponential Expressions
 Square Roots
Evaluating Exponential Expressions
Two Rules for Exponential Expressions:
Example:
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University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
Solution:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
59
CHAPTER 1 Introductory Information and Review
Additional Properties for Exponential Expressions:
Two Definitions:
Quotient Rule for Exponential Expressions:
Exponential Expressions with Bases of Products:
Exponential Expressions with Bases of Fractions:
Example:
Evaluate each of the following:
(a) 2 3
(b)
59
56
2
(c)  
5
3
Solution:
60
University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics
61
CHAPTER 1 Introductory Information and Review
Additional Example 1:
Solution:
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University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
Additional Example 2:
Solution:
MATH 1300 Fundamentals of Mathematics
63
CHAPTER 1 Introductory Information and Review
Additional Example 3:
Solution:
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University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics
65
CHAPTER 1 Introductory Information and Review
Square Roots
Definitions:
Two Rules for Square Roots:
Writing Radical Expressions in Simplest Radical Form:
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University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
Example:
Solution:
Example:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Solution:
Exponential Form:
Additional Example 1:
Solution:
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University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
Additional Example 2:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Solution:
Additional Example 3:
Solution:
70
University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics
71
Exercise Set 1.4: Exponents and Radicals
Write each of the following products instead as a base
and exponent. (For example, 6  6  62 )
1.
2.
(a) 7  7  7
(c) 8  8  8  8  8  8
(b) 10 10
(d) 3  3  3  3  3  3  3
(a) 9  9  9
(c) 5  5  5  5
(b) 4  4  4  4  4
(d) 17 17
Fill in the appropriate symbol from the set
3.
7 2
4.
 9 4 ______
 , ,   .
______ 0
13. (a) 52  56
(b) 52  56
14. (a) 38  35
(b) 38  35
15. (a)
69
62
(b)
69
6 2
16. (a)
79
75
(b)
79
7 5
17. (a)
4 7  43
48
(b)
411  43
48  45
18. (a)
812
8  84
(b)
84  89
84  81
19. (a)
7 
(b)
5 
20. (a)
 
(b)
 2  
0
5.
 8  6
6.
8
7.
10 2 ______
 10 2
8.
10 3 ______
 10 3
6
Write each of the following products instead as a base
and exponent. (Do not evaluate; simply write the base
and exponent.) No answers should contain negative
exponents.
______ 0
______ 0
5
3 6
32
4

3
2 4
4
3 5
Evaluate the following.
9.
1
(a) 3
(d) 3 1
(g)
(b) 3
(e) 3 2
(h)
(c) 3
(f) 3 3
 3  2
(i)
(k) 30
(l)
(m) 34
(n) 34
(o)
10. (a) 5
0
(b)
(d) 5
1
(e)
(g) 5 2
(h)
(j) 5 3
(k)
(m) 5 4
(n)
12. (a)
 0.5  2
 0.03 2
Rewrite each expression so that it contains positive
exponent(s) rather than negative exponent(s), and then
evaluate the expression.
3
(j) 3 0
11. (a)
72
 31
2
 5 
 5 1
 5  2
 5  3
 5  4
0
1
(b)  
5
2
1
(b)  
3
4
 3  3
 3  0
 3  4
(c) 5
21. (a) 5 1
(b) 5  2
(c) 5  3
22. (a) 3 1
(b) 3  2
(c) 3  3
23. (a) 2 3
(b) 2 5
24. (a) 7  2
(b) 10  4
0
(f) 5 1
(i)
5 2
(l)
5 3
1
25. (a)  
5
(o) 5 4
 1
(c)   
 9
2
(b)  
3
1
1
6
(b)  
5
1
1
26. (a)  
7
2
 1 
(c)   
 12 
1
27. (a) 5  2
(b)
 52
2
28. (a)
 82
(b) 8  2
University of Houston Department of Mathematics
Exercise Set 1.4: Exponents and Radicals
Evaluate the following.
2 2
(b) 6
2
23
29. (a)
28
30. (a)
42.
5 1
52
(b)
51
53
 2  
(b)
 2  
32. (a)
3  
(b)
3  
2
1 2
 5a 2b2 
44. 
2 
 6a b 
2
3 1
34. (a)
35.
36.
37.
x
3x y z 
3 4 2 3
 6x
y z
3
Write each of the following expressions in simplest
radical form or as a rational number (if appropriate).
If it is already in simplest radical form, say so.
(b)
(b)
45. (a)
 36 
46. (a)
20
47. (a)
 50 
48. (a)
19 
49. (a)
28
50. (a)
 45
51. (a)
1
2
(b)
7
(c)
18
(b)
49
(c)
 32 
(b)
14
(c)
81
16
(b)
16
49
(c)
55
(b)
72
(c)
 27 
(b)
48
(c)
500
54
(b)
 80 
(c)
60
52. (a)
120
(b)
(c)
 84 
53. (a)
1
5
 3 2
(b)  
4
(c)
2
7
54. (a)
1
3
3x y z 
3 4 2 3
 6x

y z
1
2

x x
x
7
1
2
x 2 x 3 x 4

x 4 x 1

1
k 3m2
 
k 1 m 2
 
1
2
5 3 4 2
3 4 6 1
a 4 b 3
38.

5 3 4 2
2
0
2 1
Simplify the following. No answers should contain
negative exponents.
33. (a)
 c  d 0
 3a3b6 
43.   3 2 
 2a b 
31. (a)
2
3 0
c0  d 0
1
2
1
2
3
4
c7
3 5 9
1
2
180
ab c
1
2
1
2a 4 b 3
39. 1 0 9
4 ab
5d 7 e0
40.
31 d 2 e4
41.
a 0  b0
a  b
0
1
55. (a)
56. (a)
MATH 1300 Fundamentals of Mathematics
7
4
1
6
(b)
(b)
(b)
5
9
1
10
11
9
 2 2
(c)  
5
(c)
(c)
3
11
5
2
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Exercise Set 1.4: Exponents and Radicals
57. (a)
35
58. (a)
7
2
(b)
x4 y5 z 7
(b)
2 9 5
63. (a)
3
8
(b)
3
8
(c)  3 8
64. (a)
4
81
(b)
4
81
(c)  4 81
65. (a)
6 1, 000, 000
a bc
(b)
6
1,000,000
(c)  6 1,000,000
Evaluate the following.
59. (a)
60. (a)
 5
2
 7
 6
(b)
2
 3
(b)
4
(c)
4
(c)
 2

(b)
5
32
(c)  5 32
4 1
16
(b)
4
 161
(c)  4
1
16
68. (a)
3 1
27
(b)
3
 271
(c)  3
1
27
69. (a)
5
(b)
5
1
 100,000
(b)
6
1
66. (a)
5
67. (a)
32
6
10

6
We can evaluate radicals other than square roots.
With square roots, we know, for example, that
49  7 , since 7 2  49 , and  49 is not a real
number. (There is no real number that when squared
1
100,000
(c)  5
70. (a)
6
1
1
100,000
(c)  6 1
gives a value of 49 , since 7 and  7  give a value
2
2
of 49, not 49 . The answer is a complex number,
which will not be addressed in this course.) In a
similar fashion, we can compute the following:
Cube Roots
3
125  5 , since 53  125 .
3
125  5 , since  5   125 .
3
Fourth Roots
4 10, 000  10 , since 104  10, 000 .
4
10, 000 is not a real number.
Fifth Roots
5
32  2 , since 25  32 .
5
32  2 , since  2   32 .
5
Sixth Roots
6 1
64
6
 12 , since
 12 
6
 64 .
 641 is not a real number.
Evaluate the following. If the answer is not a real
number, state “Not a real number.”
74
61. (a)
64
(b)
64
(c)  64
62. (a)
25
(b)
25
(c)  25
University of Houston Department of Mathematics