The document provides examples and definitions for properties and operations involving exponents. It defines properties like the product of powers, power of a power, quotient of powers, and zero exponents. It also defines negative integer exponents and provides examples of simplifying expressions using the definition that a^-n = 1/an.

1. Section 7-2 (and 7-3)
Properties of Powers (and Negative Integer
Exponents)

2. Warm-up
A multiple choice quiz has two questions. For each
question, there are three choices: A, B, and C. List all
possible ways for a student to complete the quiz.
Is the possible number of ways 23 or 32?

3. Warm-up
A multiple choice quiz has two questions. For each
question, there are three choices: A, B, and C. List all
possible ways for a student to complete the quiz.
(1A, 2A), (1A, 2B), (1A, 2C),
(1B, 2A), (1B, 2B), (1B, 2C),
(1C, 2A), (1C, 2B), (1C, 2C)
Is the possible number of ways 23 or 32?

4. Warm-up
A multiple choice quiz has two questions. For each
question, there are three choices: A, B, and C. List all
possible ways for a student to complete the quiz.
(1A, 2A), (1A, 2B), (1A, 2C),
(1B, 2A), (1B, 2B), (1B, 2C),
(1C, 2A), (1C, 2B), (1C, 2C)
Is the possible number of ways 23 or 32?
32

5. Example 1
2 4
3 i3

6. Example 1
3 i3 = (3i3)(3i3i3i3)
2 4

7. Example 1
3 i3 = (3i3)(3i3i3i3) = 3
2 4 6

8. Example 1
3 i3 = (3i3)(3i3i3i3) = 3
2 4 6
2 4
3 i3

9. Example 1
3 i3 = (3i3)(3i3i3i3) = 3
2 4 6
3 i3 = 3
2 4 2+4

10. Example 1
3 i3 = (3i3)(3i3i3i3) = 3
2 4 6
3 i3 = 3 =3
2 4 2+4 6

11. Product of Powers Postulate
b ib = b
m n m+ n

12. Example 2
24
(3 )

13. Example 2
(3 ) = (3 )(3 )(3 )(3 )
24 2 2 2 2

14. Example 2
(3 ) = (3 )(3 )(3 )(3 ) = 3
8
24 2 2 2 2

15. Example 2
(3 ) = (3 )(3 )(3 )(3 ) = 3 8
24 2 2 2 2
24
(3 )

16. Example 2
(3 ) = (3 )(3 )(3 )(3 ) = 3 8
24 2 2 2 2
24
(3 ) = 3 2i4

17. Example 2
(3 ) = (3 )(3 )(3 )(3 ) = 3 8
24 2 2 2 2
24
(3 ) = 3 =3
2i4 8

18. Power of a Power Postulate
(b ) = b
mn mn

19. Example 3
2 3
(x y)

20. Example 3
( x y ) = ( x y )( x y )( x y )
2 3 2 2 2

21. Example 3
( x y ) = ( x y )( x y )( x y ) = x y
2 3 2 2 2 6 3

22. Example 3
( x y ) = ( x y )( x y )( x y ) = x y
2 3 2 2 2 6 3
2 3
(x y)

23. Example 3
( x y ) = ( x y )( x y )( x y ) = x y
2 3 2 2 2 6 3
(x y) = (x ) y
2 3 23 3

24. Example 3
( x y ) = ( x y )( x y )( x y ) = x y
2 3 2 2 2 6 3
(x y) = (x ) y = x y
2 3 23 3 6 3

25. Power of a Product Postulate
(ab) = a b
m m m

26. Example 4
2 34
(3x y )

27. Example 4
(3x y ) = 3 ( x ) ( y )
2 34 4 24 34

28. Example 4
(3x y ) = 3 ( x ) ( y ) = 81x y
2 34 4 24 34 8 12

29. Example 5
11
10
8
10

30. Example 5
11
(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)
10
=
(10)(10)(10)(10)(10)(10)(10)(10)
8
10

31. Example 5
11
(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)
10
=
(10)(10)(10)(10)(10)(10)(10)(10)
8
10

32. Example 5
11
(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)
10
=
(10)(10)(10)(10)(10)(10)(10)(10)
8
10
= 10 3

33. Example 5
11
(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)
10
=
(10)(10)(10)(10)(10)(10)(10)(10)
8
10
= 10 3
11
10
8
10

34. Example 5
11
(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)
10
=
(10)(10)(10)(10)(10)(10)(10)(10)
8
10
= 10 3
11
10
= 10
11−8
8
10

35. Example 5
11
(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)
10
=
(10)(10)(10)(10)(10)(10)(10)(10)
8
10
= 10 3
11
10
= 10 = 10
11−8 3
8
10

36. Quotient of Powers Postulate
m
b
=b m− n
n
b

37. Example 6
6
⎛ x⎞
⎜ ⎟
⎝ y⎠

38. Example 6
⎛ x⎞ ⎛ x⎞ ⎛ x⎞ ⎛ x⎞ ⎛ x⎞ ⎛ x⎞
6
⎛ x⎞
=⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎜ ⎟ ⎝ y⎠ ⎝ y⎠ ⎝ y⎠ ⎝ y⎠ ⎝ y⎠ ⎝ y⎠
⎝ y⎠

39. Example 6
⎛ x⎞ ⎛ x⎞ ⎛ x⎞ ⎛ x⎞ ⎛ x⎞ ⎛ x⎞
6 6
x
⎛ x⎞
⎟= 6
=⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜
⎜ ⎟ ⎝ y⎠ ⎝ y⎠ ⎝ y⎠ ⎝ y⎠ ⎝ y⎠ ⎝ y⎠ y
⎝ y⎠

40. Power of a Quotient Postulate
m
⎛ a⎞ m
a
⎜ b⎟ = m
⎝⎠ b

41. Example 7
3
x
3
x

42. Example 7
3
x
=x 3−3
3
x

43. Example 7
3
x
=x =x
3−3 0
3
x

44. Example 7
3
x
=x =x
3−3 0
3
x
3
x
3
x

45. Example 7
3
x
=x =x
3−3 0
3
x
3
x
=1
3
x

46. Example 7
3
x
=x =x
3−3 0
3
x
3
x
=1
3
x
x =10

47. Zero Exponent Theorem
b =1 b≠ 0
0
,

48. 7-3: Negative Integer
Exponents

49. Example 1
7
x
10
x

50. Example 1
7
x
=x 7−10
10
x

51. Example 1
7
x −3
=x =x
7−10
10
x

52. Example 1
7
x −3
=x =x
7−10
10
x
7
x
10
x

53. Example 1
7
x −3
=x =x
7−10
10
x
7
( x)( x)( x)( x)( x)( x)( x)
x
=
( x)( x)( x)( x)( x)( x)( x)( x)( x)( x)
10
x

54. Example 1
7
x −3
=x =x
7−10
10
x
7
( x)( x)( x)( x)( x)( x)( x)
x
=
( x)( x)( x)( x)( x)( x)( x)( x)( x)( x)
10
x
1
=
( x)( x)( x)

55. Example 1
7
x −3
=x =x
7−10
10
x
7
( x)( x)( x)( x)( x)( x)( x)
x
=
( x)( x)( x)( x)( x)( x)( x)( x)( x)( x)
10
x
1 1
= =3
( x)( x)( x) x

56. Negative Exponent Theorem
1
−n
=
b n
b

57. Example 2
−5
3
(5b) (4b)

58. Example 2
−5
3
(5b) (4b)
3
(5b)
= 5
(4b)

59. Example 2
−5
3
(5b) (4b)
3 3
(5b) 125b
= =
5
(4b) 5
1024b

60. Example 2
−5
3
(5b) (4b)
3 3
(5b) 125b
= =
5
(4b) 5
1024b
125
= 2
1024b

61. Homework

62. Homework
p. 430 #14-30
p. 435 #9-21, 25