WEBVTT 00:00:00.740 --> 00:00:05.450 I have been asked for some intuition as to why, let's say, 00:00:05.450 --> 00:00:12.030 a to the minus b is equal to one over a to the b. 00:00:12.030 --> 00:00:13.382 And before I give you the intuition, 00:00:13.382 --> 00:00:17.420 I want you to just realize that this really is a definition. 00:00:17.420 --> 00:00:17.920 I don't know. 00:00:17.920 --> 00:00:20.950 The inventor of mathematics wasn't one person. 00:00:20.950 --> 00:00:23.120 It was, you know, a convention that arose. 00:00:23.120 --> 00:00:25.180 But they defined this, 00:00:25.180 --> 00:00:28.634 and they defined this for the reasons that I'm going to show you. 00:00:28.634 --> 00:00:30.477 Well, what I'm going to show you is one of the reasons, 00:00:30.477 --> 00:00:32.593 and then we'll see that this is a good definition, 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 00:00:38.790 --> 00:00:41.596 and when you raise something to the zeroth power. 00:00:41.596 --> 00:00:44.740 So let's take the positive exponents. 00:00:44.740 --> 00:00:47.180 Those are pretty intuitive, I think. 00:00:47.180 --> 00:00:54.200 So the positive exponents, so you have a to the one, a squared, 00:00:54.200 --> 00:00:58.140 a cubed, a to the fourth. 00:00:58.140 --> 00:01:01.832 What's a to the one? a to the one, we said, was a, 00:01:01.832 --> 00:01:06.060 and then to get to a squared, what did we do? 00:01:06.060 --> 00:01:08.200 We multiplied by a, right? 00:01:08.200 --> 00:01:10.650 a squared is just a times a. 00:01:10.650 --> 00:01:13.040 And then to get to a cubed, what did we do? 00:01:13.040 --> 00:01:15.160 We multiplied by a again. 00:01:15.160 --> 00:01:17.420 And then to get to a to the fourth, what did we do? 00:01:17.420 --> 00:01:18.920 We multiplied by a again. 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? 00:01:24.480 --> 00:01:29.560 We are multiplying by one over a, or dividing by a. 00:01:29.560 --> 00:01:33.140 And similarly, you decrease again, you're dividing by a. 00:01:33.140 --> 00:01:38.479 And to go from a squared to a to the first, you're dividing by a. 00:01:38.479 --> 00:01:41.700 So let's use this progression to figure out what a to the zero is. 00:01:41.720 --> 00:01:43.900 So this is the first hard one. 00:01:43.900 --> 00:01:45.010 So a to the zero. 00:01:45.010 --> 00:01:49.990 So you're the inventor, the founding mother of mathematics, 00:01:49.990 --> 00:01:52.170 and you need to define what a to the zero is. 00:01:52.170 --> 00:01:55.420 And, you know, maybe it's seventeen, maybe it's pi. 00:01:55.420 --> 00:01:56.100 I don't know. 00:01:56.100 --> 00:01:58.860 It's up to you to decide what a to the zero is. 00:01:58.860 --> 00:02:02.140 But wouldn't it be nice if a to the zero retained this pattern? 00:02:02.140 --> 00:02:07.274 That every time you decrease the exponent, you're dividing by a, right? 00:02:07.274 --> 00:02:11.700 So, if you're going from a to the first to a to the zero, 00:02:11.700 --> 00:02:14.160 wouldn't it be nice if we just divided by a? 00:02:14.160 --> 00:02:15.189 So let's do that. 00:02:15.189 --> 00:02:18.320 So if we go from a to the first, which is just a, 00:02:18.320 --> 00:02:21.078 and divide by a, 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. 00:02:23.863 --> 00:02:27.235 What is a divided by a? 00:02:27.235 --> 00:02:29.730 Well, it's just one. 00:02:29.730 --> 00:02:30.994 So that's where the definition-- 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. 00:02:37.420 --> 00:02:39.456 Because when you take that number 00:02:39.456 --> 00:02:43.190 and divide it by itself one more time, you just get one. 00:02:43.190 --> 00:02:44.177 So that's pretty reasonable, 00:02:44.177 --> 00:02:45.890 but now let's go into the negative domain. 00:02:45.890 --> 00:02:51.891 So what should a to the negative one equal? 00:02:51.891 --> 00:02:54.410 Well, once again, it's nice if we can retain this pattern, 00:02:54.410 --> 00:02:57.682 where every time we decrease the exponent we're dividing by a. 00:02:57.682 --> 00:03:01.546 So let's divide by a again, so one over a. 00:03:01.546 --> 00:03:06.140 So we're going to take a to the zero and divide it by a. 00:03:06.140 --> 00:03:09.610 a to the zero is one, so what's one divided by a? 00:03:09.610 --> 00:03:12.090 It's one over a. 00:03:12.090 --> 00:03:13.078 Now, let's do it one more time, 00:03:13.078 --> 00:03:15.330 and then I think you're going to get the pattern. 00:03:15.330 --> 00:03:16.880 Well, I think you probably already got the pattern. 00:03:16.880 --> 00:03:18.350 What's a to the minus two? 00:03:18.350 --> 00:03:21.993 Well, we want-- you know, it'd be silly now to change this pattern. 00:03:21.993 --> 00:03:25.130 Every time we decrease the exponent, we're dividing by a. 00:03:25.130 --> 00:03:27.840 So to go from a to the minus one to a to the minus two, 00:03:27.855 --> 00:03:30.470 let's just divide by a again. 00:03:30.470 --> 00:03:32.550 And what do we get? 00:03:32.550 --> 00:03:36.040 If you take one over a and divide by a, you get one over a squared. 00:03:36.040 --> 00:03:39.146 And you could just keep doing this pattern all the way to the left, 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. 00:03:44.761 --> 00:03:48.790 Hopefully, that gave you a little intuition as to why-- 00:03:48.790 --> 00:03:51.090 well, first of all, you know, the big mystery is, you know, 00:03:51.090 --> 00:03:53.590 something to the zeroth power, why does that equal one? 00:03:53.590 --> 00:03:55.970 First, keep in mind that that's just a definition. 00:03:55.972 --> 00:03:59.134 Someone decided that it should be equal to one, but they had a good reason. 00:03:59.134 --> 00:04:02.617 And their good reason was they wanted to keep this pattern going. 00:04:02.617 --> 00:04:07.422 And that's the same reason why they defined negative exponents in this way. 00:04:07.440 --> 00:04:08.654 And what's extra cool about it is 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, 00:04:13.227 --> 00:04:16.138 or when you're increasing exponents, you're multiplying by a, 00:04:16.138 --> 00:04:20.457 but as you'll see in the exponent rules videos, all of the exponent rules hold. 00:04:20.460 --> 00:04:25.574 All of the exponent rules are consistent with this definition of something to the zeroth power 00:04:25.574 --> 00:04:28.472 and this definition of something to the negative power. 00:04:28.472 --> 00:04:30.290 Hopefully, that didn't confuse you 00:04:30.290 --> 00:04:34.010 and gave you a little bit of intuition and demystified something that, frankly, 00:04:34.010 --> 00:04:37.545 is quite mystifying the first time you learn it.