Patterns in raising 1 and -1 to different powers | Pre-Algebra | Khan Academy — Transcript

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1. 0:00Let's think about exponents with ones and zeroes.
2. 0:04So let's take the number 1, and let's
3. 0:07raise it to the eighth power.
4. 0:10So we've already seen that there's
5. 0:11two ways of thinking about this.
6. 0:12You could literally view this as taking eight 1's, and then
7. 0:15multiplying them together.
8. 0:16So let's do that.
9. 0:17So you have one, two, three, four, five, six, seven,
10. 0:21eight 1's, and then you're going to multiply them together.
11. 0:26And if you were to do that, you would get well, 1 times 1
12. 0:29is 1, times 1-- it doesn't matter
13. 0:31how many times you multiply 1 by 1.
14. 0:33You are going to just get 1.
15. 0:36You are just going to get 1.
16. 0:38And you could imagine.
17. 0:39I did it eight times.
18. 0:40I multiplied eight 1's.
19. 0:41But even if this was 80, or if this was 800,
20. 0:44or if this was 8 million, if I just multiplied 1--
21. 0:46if I had 8 million 1's, and I multiplied them all together,
22. 0:50it would still be equal to 1.
23. 0:53So 1 to any power is just going to be equal to 1.
24. 0:57And you might say, hey, what about 1 to the 0 power?
25. 1:02Well, we've already said anything to 0 power,
26. 1:05except for 0-- that's where we're
27. 1:07going to-- it's actually up for debate.
28. 1:09But anything to the 0 power is going to be equal to 1.
29. 1:13And just as a little bit of intuition
30. 1:14here, you could literally view this
31. 1:16as our other definition of exponentiation, which
32. 1:19is you start with a 1, and this number
33. 1:22says how many times you're going to multiply that 1 times
34. 1:24this number.
35. 1:25So 1 times 1 zero times is just going to be 1.
36. 1:30And that was a little bit clearer when we did it
37. 1:32like this, where we said 2 to the,
38. 1:34let's say, fourth power is equal to-- this
39. 1:39was the other definition of exponentiation we had,
40. 1:42which is you start with a 1, and then you multiply it
41. 1:45by 2 four times, so times 2, times 2, times 2,
42. 1:50times 2, which is equal to-- let's see, this is equal to 16.
43. 1:56So here if you start with a 1 and then
44. 1:59you multiply it by 1 zero times, you're
45. 2:02still going to have that 1 right over there.
46. 2:03And that's why anything that's not 0 to the 1 power
47. 2:07is going to be equal to 1.
48. 2:09Now let's try some other interesting scenarios.
49. 2:12Let's start try some negative numbers.
50. 2:14So let's take negative 1.
51. 2:17And let's first raise it to the 0 power.
52. 2:22So once again, this is just going,
53. 2:25based on this definition, this is starting with a 1
54. 2:28and then multiplying it by this number 0 times.
55. 2:30Well, that means we're just not going
56. 2:32to multiply it by this number.
57. 2:33So you're just going to get a 1.
58. 2:35Let's try negative 1.
59. 2:37Let's try negative 1 to the first power.
60. 2:41Well, anything to the first power, you could view this--
61. 2:43and I like going with this definition as opposed
62. 2:45to this one right over here.
63. 2:46If we were to make them consistent,
64. 2:48if you were to make this definition
65. 2:49consistent with this, you would say hey, let's start with a 1,
66. 2:52and then multiply it by 1 eight times.
67. 2:56And you're still going to get a 1 right over here.
68. 2:58But let's do this with negative 1.
69. 3:00So we're going to start with a 1,
70. 3:02and then we're going to multiply it by negative 1 one time--
71. 3:06times negative 1.
72. 3:07And this is, of course, going to be equal to negative 1.
73. 3:11Now let's take negative 1, and let's
74. 3:14take it to the second power.
75. 3:15We often say that we are squaring it
76. 3:16when we take something to the second power.
77. 3:18So negative 1 to the second power-- well,
78. 3:21we could start with a 1.
79. 3:23We could start with a 1, and then multiply it by negative 1
80. 3:26two times-- multiply it by negative 1 twice.
81. 3:32And what's this going to be equal to?
82. 3:34And once again, by our old definition,
83. 3:35you could also just say, hey, ignoring this one,
84. 3:38because that's not going to change the value,
85. 3:39we took two negative 1's and we're multiplying them.
86. 3:42Well, negative 1 times negative 1 is 1.
87. 3:45And I think you see a pattern forming.
88. 3:48Let's take negative 1 to the third power.
89. 3:54What's this going to be equal to?
90. 3:56Well, by this definition, you start with a 1,
91. 3:58and then you multiply it by negative 1 three times,
92. 4:01so negative 1 times negative 1 times negative 1.
93. 4:06Or you could just think of it as you're
94. 4:08taking three negative 1's and you're multiplying it,
95. 4:10because this 1 doesn't change the value.
96. 4:12And this is going to be equal to negative 1 times negative 1
97. 4:15is positive 1, times negative 1 is negative 1.
98. 4:19So you see the pattern.
99. 4:20Negative 1 to the 0 power is 1.
100. 4:22Negative 1 to the first power is negative 1.
101. 4:24Then you multiply it by negative 1,
102. 4:25you're going to get positive 1.
103. 4:27Then you multiply it by negative 1 again to get negative 1.
104. 4:30And the pattern you might be seeing
105. 4:31is if you take negative 1 to an odd power you're
106. 4:38going to get negative 1.
107. 4:40And if you take it to an even power,
108. 4:42you're going to get 1 because a negative times a negative is
109. 4:46going to be the positive.
110. 4:47And you're going to have an even number of negatives,
111. 4:50so that you're always going to have negative times negatives.
112. 4:52So this right over here, this is even.
113. 4:56Even is going to be positive 1.
114. 4:57And then you could see that if you went to negative 1
115. 5:00to the fourth power.
116. 5:01Negative 1 the fourth power?
117. 5:04Well, you could start with a 1 and then
118. 5:06multiply it by negative 1 four times, so a negative 1 times
119. 5:11negative 1, times negative 1, times negative 1,
120. 5:15which is just going to be equal to positive 1.
121. 5:18So if someone were to ask you-- we already
122. 5:21established that if someone were to take 1 to the, I don't know,
123. 5:261 millionth power, this is just going to be equal to 1.
124. 5:35If someone told you let's take negative 1
125. 5:38and raise it to the 1 millionth power, well, 1 million
126. 5:44is an even number, so this is still
127. 5:46going to be equal to positive 1.
128. 5:49But if you took negative 1 to the 999,999th power,
129. 5:58this is an odd number.
130. 6:00So this is going to be equal to negative 1.