Raising a number to the 0 and 1st power | Pre-Algebra | Khan Academy — Transcript

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1. 0:00What I want to do in this video is
2. 0:02think about exponents in a slightly different way that
3. 0:04will be useful for different contexts
4. 0:06and also go through a lot more examples.
5. 0:09So in the last video, we saw that taking something
6. 0:11to an exponent means multiplying that number that many times.
7. 0:14So if I had the number negative 2
8. 0:17and I want to raise it to the third power,
9. 0:20this literally means taking three negative 2's,
10. 0:26so negative 2, negative 2, and negative 2,
11. 0:30and then multiplying them.
12. 0:34So what's this going to be?
13. 0:35Well, let's see.
14. 0:36Negative 2 times negative 2 is positive 4,
15. 0:40and then positive 4 times negative 2 is negative 8.
16. 0:45So this would be equal to negative 8.
17. 0:51Now, another way of thinking about exponents,
18. 0:53instead of saying you're just taking three negative 2's
19. 0:55and multiplying them, and this is a completely reasonable way
20. 0:58of viewing it, you could also view it
21. 0:59as this is a number of times you're
22. 1:02going to multiply this number times 1.
23. 1:05So you could completely view this
24. 1:07as being equal to-- so you're going to start with a 1,
25. 1:10and you're going to multiply 1 times negative 2 three times.
26. 1:15So this is times negative 2 times negative 2
27. 1:22times negative 2.
28. 1:25So clearly these are the same number.
29. 1:27Here we just took this, and we're just multiplying it by 1,
30. 1:29so you're still going to get negative 8.
31. 1:33And this might be a slightly more useful idea
32. 1:35to get an intuition for exponents, especially when you
33. 1:38start taking things to the 1 or 0 power.
34. 1:42So let's think about that a little bit.
35. 1:44What is positive 2 to the-- based
36. 1:49on this definition-- to the 0 power going to be equal to?
37. 1:58Well, we just said.
38. 1:59This says how many times are going to multiply 1 times
39. 2:02this number?
40. 2:03So this literally says, I'm going to take a 1,
41. 2:08and I'm going to multiply by 2 zero times.
42. 2:11Well, if I want to multiply it by 2 zero times,
43. 2:13that means I'm just left with the 1.
44. 2:15So 2 to the zero power is going to be equal to 1.
45. 2:18And, actually, any non-zero number to the 0 power
46. 2:21is 1 by that same rationale.
47. 2:23And I'll make another video that will also
48. 2:25give a little bit more intuition on there.
49. 2:27That might seem very counterintuitive,
50. 2:29but it's based on one way of thinking
51. 2:31about it is thinking of an exponent as this.
52. 2:33And this will also make sense if we
53. 2:35start thinking of what 2 to the first power is.
54. 2:41So let's go to this definition we just gave of the exponent.
55. 2:44We always start with a 1, and we multiply it by the 2 one time.
56. 2:51So 2 is going to be 1-- we're only
57. 2:53going to multiply it by the 2.
58. 2:54I'll use this for multiplication.
59. 2:56I'll use the dot.
60. 2:57We're only going to multiply it by 2 one time.
61. 3:01So 1 times 2, well, that's clearly
62. 3:03just going to be equal to 2.
63. 3:06And any number to the first power
64. 3:08is just going to be equal to that number.
65. 3:11And then we can go from there, and you will, of course,
66. 3:13see the pattern.
67. 3:14If we say what 2 squared is, well, based on this definition,
68. 3:21we start with a 1, and we multiply it by 2 two times.
69. 3:27So times 2 times 2 is going to be equal to 4.
70. 3:34And we've seen this before.
71. 3:35You go to 2 to the third, you start with the 1,
72. 3:43and then multiply it by 2 three times.
73. 3:47So times 2 times 2 times 2.
74. 3:51This is going to give us positive 8.
75. 3:54And you probably see a pattern here.
76. 3:56Every time we multiply by 2-- or every time, I should say,
77. 4:00we raise 2 to one more power, we are multiplying by 2.
78. 4:04Notice this, to go from 2 to the 0 to 2 to the 1,
79. 4:07we multiplied by 2.
80. 4:10I'll use a little x for the multiplication symbol now,
81. 4:13a little cross.
82. 4:14And then to go from 2 to the first power to 2
83. 4:17to the second power, we multiply by 2 and multiply by 2 again.
84. 4:21And that makes complete sense because this is literally
85. 4:23telling us how many times are we going to take this number
86. 4:26and-- how many times are we going take 1 and multiply it
87. 4:29by this number?
88. 4:30And so when you go from 2 to the second power to 2 to the third,
89. 4:33you're multiplying by 2 one more time.
90. 4:37And this is another intuition of why something to the 0 power
91. 4:40is equal to 1.
92. 4:41If you were to go backwards, if, say,
93. 4:43we didn't know what 2 to the 0 power
94. 4:45is and we were just trying to figure out
95. 4:46what would make sense, well, when
96. 4:47we go from 2 to the third power to 2 to the second,
97. 4:50we'd be dividing by 2.
98. 4:51We're going from 8 to 4.
99. 4:53Then we'd divide by 2 again to go from 2 to the second to 2
100. 4:56to the first.
101. 4:57And then it seems like we should just
102. 4:59divide by 2 again from going from 2 to the first
103. 5:01to 2 to the 0.
104. 5:02And that would give us 1.