Negative exponent intuition

I have been asked for some intuition as to why, let's say, a to the minus b is equal to 1 over a to the b. And before I give you the intuition, I want you to just realize that this really is a definition. I don't know. The inventor of mathematics wasn't one person. It was, you know, a convention that arose. But they defined this, and they defined this for the reasons that I'm going to show you. Well, what I'm going to show you is one of the reasons, and then we'll see that this is a good definition, because once you learned exponent rules, all of the other exponent rules stay consistent for negative exponents and when you raise something to the zero power. So let's take the positive exponents. Those are pretty intuitive, I think. So the positive exponents, so you have a to the 1, a squared, a cubed, a to the fourth. What's a to the 1? a to the 1, we said, is a, and then to get to a squared, what did we do? We multiplied by a, right? a squared is just a times a. And then to get to a cubed, what did we do? We multiplied by a again. And then to get to a to the fourth, what did we do? We multiplied by a again. Or the other way, you could imagine, is when you decrease the exponent, what are we doing? We are multiplying by 1/a, or dividing by a. And similarly, you decrease again, you're dividing by a. And to go from a squared to a to the first, you're dividing by a. So let's use this progression to figure out what a to the 0 is. So this is the first hard one. So a to the 0. So you're the inventor, the founding mother of mathematics, and you need to define what a to the 0 is. And, you know, maybe it's 17, maybe it's pi. I don't know. It's up to you to decide what a to the 0 is. But wouldn't it be nice if a to the 0 retained this pattern? That every time you decrease the exponent, you're dividing by a, right? So if you're going from a to the first to a to the zero, wouldn't it be nice if we just divided by a? So let's do that. So if we go from a to the first, which is just a, and divide by a, right, so we're just going to go-- we're just going to divide it by a, what is a divided by a? Well, it's just 1. So that's where the definition-- or that's one of the intuitions behind why something to the 0-th power is equal to 1. Because when you take that number and you divide it by itself one more time, you just get 1. So that's pretty reasonable, but now let's go into the negative domain. So what should a to the negative 1 equal? Well, once again, it's nice if we can retain this pattern, where every time we decrease the exponent we're dividing by a. So let's divide by a again, so 1/a. So we're going to take a to the 0 and divide it by a. a to the 0 is one, so what's 1 divided by a? It's 1/a. Now, let's do it one more time, and then I think you're going to get the pattern. Well, I think you probably already got the pattern. What's a to the minus 2? Well, we want-- you know, it'd be silly now to change this pattern. Every time we decrease the exponent, we're dividing by a, so to go from a to the minus 1 to a to the minus 2, let's just divide by a again. And what do we get? If you take 1/2 and divide by a, you get 1 over a squared. And you could just keep doing this pattern all the way to the left, and you would get a to the minus b is equal to 1 over a to the b. Hopefully, that gave you a little intuition as to why-- well, first of all, you know, the big mystery is, you know, something to the 0-th power, why does that equal 1? First, keep in mind that that's just a definition. Someone decided it should be equal to 1, but they had a good reason. And their good reason was they wanted to keep this pattern going. And that's the same reason why they defined negative exponents in this way. And what's extra cool about it is not only does it retain this pattern of when you decrease exponents, you're dividing by a, or when you're increasing exponents, you're multiplying by a, but as you'll see in the exponent rules videos, all of the exponent rules hold. All of the exponent rules are consistent with this definition of something to the 0-th power and this definition of something to the negative power. Hopefully, that didn't confuse you and gave you a little bit of intuition and demystified something that, frankly, is quite mystifying the first time you learn it.