1 00:00:00,740 --> 00:00:05,450 I have been asked for some intuition as to why, let's say, 2 00:00:05,450 --> 00:00:12,030 a to the minus b is equal to one over a to the b. 3 00:00:12,030 --> 00:00:13,382 And before I give you the intuition, 4 00:00:13,382 --> 00:00:17,420 I want you to just realize that this really is a definition. 5 00:00:17,420 --> 00:00:17,920 I don't know. 6 00:00:17,920 --> 00:00:20,950 The inventor of mathematics wasn't one person. 7 00:00:20,950 --> 00:00:23,120 It was, you know, a convention that arose. 8 00:00:23,120 --> 00:00:25,180 But they defined this, 9 00:00:25,180 --> 00:00:28,634 and they defined this for the reasons that I'm going to show you. 10 00:00:28,634 --> 00:00:30,477 Well, what I'm going to show you is one of the reasons, 11 00:00:30,477 --> 00:00:32,593 and then we'll see that this is a good definition, 12 00:00:32,593 --> 00:00:38,790 because once you learn the exponent rules, all of the other exponent rules stay consistent for negative exponents 13 00:00:38,790 --> 00:00:41,596 and when you raise something to the zeroth power. 14 00:00:41,596 --> 00:00:44,740 So let's take the positive exponents. 15 00:00:44,740 --> 00:00:47,180 Those are pretty intuitive, I think. 16 00:00:47,180 --> 00:00:54,200 So the positive exponents, so you have a to the one, a squared, 17 00:00:54,200 --> 00:00:58,140 a cubed, a to the fourth. 18 00:00:58,140 --> 00:01:01,832 What's a to the one? a to the one, we said, was a, 19 00:01:01,832 --> 00:01:06,060 and then to get to a squared, what did we do? 20 00:01:06,060 --> 00:01:08,200 We multiplied by a, right? 21 00:01:08,200 --> 00:01:10,650 a squared is just a times a. 22 00:01:10,650 --> 00:01:13,040 And then to get to a cubed, what did we do? 23 00:01:13,040 --> 00:01:15,160 We multiplied by a again. 24 00:01:15,160 --> 00:01:17,420 And then to get to a to the fourth, what did we do? 25 00:01:17,420 --> 00:01:18,920 We multiplied by a again. 26 00:01:18,920 --> 00:01:24,480 Or the other way, you could imagine, is when you decrease the exponent, what are we doing? 27 00:01:24,480 --> 00:01:29,560 We are multiplying by one over a, or dividing by a. 28 00:01:29,560 --> 00:01:33,140 And similarly, you decrease again, you're dividing by a. 29 00:01:33,140 --> 00:01:38,479 And to go from a squared to a to the first, you're dividing by a. 30 00:01:38,479 --> 00:01:41,700 So let's use this progression to figure out what a to the zero is. 31 00:01:41,720 --> 00:01:43,900 So this is the first hard one. 32 00:01:43,900 --> 00:01:45,010 So a to the zero. 33 00:01:45,010 --> 00:01:49,990 So you're the inventor, the founding mother of mathematics, 34 00:01:49,990 --> 00:01:52,170 and you need to define what a to the zero is. 35 00:01:52,170 --> 00:01:55,420 And, you know, maybe it's seventeen, maybe it's pi. 36 00:01:55,420 --> 00:01:56,100 I don't know. 37 00:01:56,100 --> 00:01:58,860 It's up to you to decide what a to the zero is. 38 00:01:58,860 --> 00:02:02,140 But wouldn't it be nice if a to the zero retained this pattern? 39 00:02:02,140 --> 00:02:07,274 That every time you decrease the exponent, you're dividing by a, right? 40 00:02:07,274 --> 00:02:11,700 So, if you're going from a to the first to a to the zero, 41 00:02:11,700 --> 00:02:14,160 wouldn't it be nice if we just divided by a? 42 00:02:14,160 --> 00:02:15,189 So let's do that. 43 00:02:15,189 --> 00:02:18,320 So if we go from a to the first, which is just a, 44 00:02:18,320 --> 00:02:21,078 and divide by a, 45 00:02:21,078 --> 00:02:23,848 right, so we're just going to go-- we're just going to divide it by a. 46 00:02:23,863 --> 00:02:27,235 What is a divided by a? 47 00:02:27,235 --> 00:02:29,730 Well, it's just one. 48 00:02:29,730 --> 00:02:30,994 So that's where the definition-- 49 00:02:30,994 --> 00:02:37,420 or that's one of the intuitions behind why something to the zeroth power is equal to one. 50 00:02:37,420 --> 00:02:39,456 Because when you take that number 51 00:02:39,456 --> 00:02:43,190 and divide it by itself one more time, you just get one. 52 00:02:43,190 --> 00:02:44,177 So that's pretty reasonable, 53 00:02:44,177 --> 00:02:45,890 but now let's go into the negative domain. 54 00:02:45,890 --> 00:02:51,891 So what should a to the negative one equal? 55 00:02:51,891 --> 00:02:54,410 Well, once again, it's nice if we can retain this pattern, 56 00:02:54,410 --> 00:02:57,682 where every time we decrease the exponent we're dividing by a. 57 00:02:57,682 --> 00:03:01,546 So let's divide by a again, so one over a. 58 00:03:01,546 --> 00:03:06,140 So we're going to take a to the zero and divide it by a. 59 00:03:06,140 --> 00:03:09,610 a to the zero is one, so what's one divided by a? 60 00:03:09,610 --> 00:03:12,090 It's one over a. 61 00:03:12,090 --> 00:03:13,078 Now, let's do it one more time, 62 00:03:13,078 --> 00:03:15,330 and then I think you're going to get the pattern. 63 00:03:15,330 --> 00:03:16,880 Well, I think you probably already got the pattern. 64 00:03:16,880 --> 00:03:18,350 What's a to the minus two? 65 00:03:18,350 --> 00:03:21,993 Well, we want-- you know, it'd be silly now to change this pattern. 66 00:03:21,993 --> 00:03:25,130 Every time we decrease the exponent, we're dividing by a. 67 00:03:25,130 --> 00:03:27,840 So to go from a to the minus one to a to the minus two, 68 00:03:27,855 --> 00:03:30,470 let's just divide by a again. 69 00:03:30,470 --> 00:03:32,550 And what do we get? 70 00:03:32,550 --> 00:03:36,040 If you take one over a and divide by a, you get one over a squared. 71 00:03:36,040 --> 00:03:39,146 And you could just keep doing this pattern all the way to the left, 72 00:03:39,146 --> 00:03:44,761 and you would get a to the minus b is equal to one over a to the b. 73 00:03:44,761 --> 00:03:48,790 Hopefully, that gave you a little intuition as to why-- 74 00:03:48,790 --> 00:03:51,090 well, first of all, you know, the big mystery is, you know, 75 00:03:51,090 --> 00:03:53,590 something to the zeroth power, why does that equal one? 76 00:03:53,590 --> 00:03:55,970 First, keep in mind that that's just a definition. 77 00:03:55,972 --> 00:03:59,134 Someone decided that it should be equal to one, but they had a good reason. 78 00:03:59,134 --> 00:04:02,617 And their good reason was they wanted to keep this pattern going. 79 00:04:02,617 --> 00:04:07,422 And that's the same reason why they defined negative exponents in this way. 80 00:04:07,440 --> 00:04:08,654 And what's extra cool about it is 81 00:04:08,654 --> 00:04:13,227 not only does it retain this pattern of when you decrease exponents, you're dividing by a, 82 00:04:13,227 --> 00:04:16,138 or when you're increasing exponents, you're multiplying by a, 83 00:04:16,138 --> 00:04:20,457 but as you'll see in the exponent rules videos, all of the exponent rules hold. 84 00:04:20,460 --> 00:04:25,574 All of the exponent rules are consistent with this definition of something to the zeroth power 85 00:04:25,574 --> 00:04:28,472 and this definition of something to the negative power. 86 00:04:28,472 --> 00:04:30,290 Hopefully, that didn't confuse you 87 00:04:30,290 --> 00:04:34,010 and gave you a little bit of intuition and demystified something that, frankly, 88 00:04:34,010 --> 00:04:37,545 is quite mystifying the first time you learn it.