Learn about Negative Exponent Intuition

Khan Academy Presents: Intuition on why a^-b = 1/(a^b) (and why a^0 =1)

I have been asked for some intuition as to why let say a(-b) is equal to 1/8(b). And before I give you the intuition I want you to just realize that this is really is the definition. I don’t know, the inventor of mathematics, you know, wasn’t one person it was a convention that a rows but they define this and they define this for the reasons that I'm going to show you. 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 learn exponent rules all of the other exponent rules stay consistent when you for negative exponents and when you raise something to the zero power. So let's take the positive exponents and those are pretty intuitive I think. So the positive exponents so you have a(1), a², a³, a(4). What's a(1)? a(1) we said was a. And then to get to a² what do we do? We multiply it by a, right a² is just a times a. And then to get to a³ would we do? We multiply it by a again. And then to get a(4) what would we do? We multiply it by a again or the other way you could imagine is when you decrease the exponent what are we doing? We are multiplying by one over a or dividing by a. And similarly decrease again you dividing by a. Let me go from a² to a(1) you dividing by a. So let use this progression and to figure out what a(0) is. So this is the first hard one. So a(0) so you're the inventor the founding mother of mathematics and you need to define what a(0) is and you know maybe it 17, maybe its some, you know, maybe its pi I don’t know its up to you to decide what a(0) is. But would it be nice if a(0) retain this pattern that every time you decrease the exponent you dividing by a. right? So if you're going from a(1) to a(0) would it be nice if we just divide it by a? So let's do that. So if we go from a(1) which is just a and divide by a, all right so we just going to divide it by a. What is a ÷ a? Well it just one, so that’s where the definition or that’s one of the intuitions behind why something to the zero power is equal to one because when you take that number and you divide it by itself one more time you just get one. So that’s pretty reasonable, But now let's go on to the negative domain. So what should a(-1) equal? Once again its 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 zero and divide it by a. a to the zero is one so what's 1 ÷ a? Its 1/a. And let's do it one more time and then I think you're going to get the pattern or I think you probably already got the pattern. What's a(-2)? Well we want to, you know, it be silly now to change this pattern every time we decrease the exponent we’re dividing by a so to go from a(-1) to the a(-2) let's just divide by a again and what do we get, we get if you take 1/a and divide by a you get 1/a². And you could just keep doing this pattern all the way to the left and you would get a(-b) is equal to 1/a(b). Hopefully that gave you a little intuition as to why—well first of all and you know the big mystery is something to a zero power why is that equal one. First keep in mind that that just the definition someone decided that it should be equal but they had a good reason. And the good reason was they wanted to keep this pattern going and that’s the same reason why they define the 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 your dividing by a or when you're increasing exponents your 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 zero of 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.