Powers of zero | Exponents, radicals, and scientific notation | Pre-algebra | Khan Academy — Transcript

Full transcript

1. 0:00In this video we're going to talk about
2. 0:01powers of
3. 0:04zero. And just as a little bit of a
4. 0:06reminder, let's start with a non-zero
5. 0:08number just to remind ourselves what
6. 0:10exponentiation is all about.
7. 0:13So if I were to take two to the first
8. 0:16power,
9. 0:17one way to think about this is we always
10. 0:19start with a one and then we multiply
11. 0:22this base that many times times that
12. 0:25one. So here we're only going to have
13. 0:26one two, so it's going to be one times
14. 0:29two, which is of course equal to two. If
15. 0:32I were to say what is two to the second
16. 0:34power, well that's going to be equal to
17. 0:38one
18. 0:39times and now I'm going to have two
19. 0:41twos, so times two times two, which is
20. 0:45equal to four. And you can keep going
21. 0:48like that. Now, the reason why I have
22. 0:51this one here, and we've done this
23. 0:52before, is to justify, and there's many
24. 0:55other good reasons why two to the zero
25. 0:58power should be equal to one. But you
26. 1:00could see if we use the same exact idea
27. 1:02here, you start with a one and then you
28. 1:04multiply it by two zero times. Well,
29. 1:07that's just going to end up with a one.
30. 1:11So so far I've told you this video is
31. 1:12about powers of zero, but I've been
32. 1:14doing powers of two. So let's focus on
33. 1:16zero now.
34. 1:17So what do you think zero to the first
35. 1:20power is going to be? Pause this video
36. 1:22and try to figure that out.
37. 1:24Well, you do the exact same idea. You
38. 1:26start with a one and then multiply it by
39. 1:29zero one time.
40. 1:31So times zero and this is going to be
41. 1:34equal to zero. What do you think zero to
42. 1:37the second power is going to be equal
43. 1:39to? Pause this video and think about
44. 1:40that.
45. 1:42Well, it's going to be one times zero
46. 1:45twice. So times zero times zero and I
47. 1:50think you see where this is going. This
48. 1:52is also going to be equal to zero.
49. 1:55What do you think zero
50. 1:58to
51. 1:59some arbitrary positive
52. 2:02positive
53. 2:04integer
54. 2:05is going to be?
55. 2:08Well, it's going to be equal to one
56. 2:10times zero that positive integer number
57. 2:13of times. So, once again, it's going to
58. 2:15be equal to zero. And in general, you
59. 2:18can extend that zero to any positive
60. 2:21value exponent is going to give you
61. 2:24zero. So, that's pretty straightforward.
62. 2:27But, there is an interesting edge case
63. 2:28here. What do you think zero to the
64. 2:32zeroth power should be? Pause this video
65. 2:34and think about that.
66. 2:37Well, this is actually contested.
67. 2:40Different people will tell you different
68. 2:41things. If you use the intuition behind
69. 2:45exponentiation that we've been using in
70. 2:47this video, you would say, "All right, I
71. 2:49would start with a one and then multiply
72. 2:52it by zero zero times." Or in other
73. 2:55words, I just wouldn't multiply it by
74. 2:56zero, in which case I'm just left with
75. 2:59the one. That zero to the zeroth power
76. 3:01should be equal to one. Other folks
77. 3:04would say, "Hey, no, I'm with the zero
78. 3:06and that's to the zeroth power, maybe it
79. 3:07should be a zero." And that's why a lot
80. 3:09of folks leave it undefined. Most of the
81. 3:11time, you're going to see zero to the
82. 3:13zeroth power either being undefined or
83. 3:17that it is equal to one.