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### Course: Pre-algebra>Unit 11

Lesson 3: Negative exponents

# Negative exponent intuition

How do negative exponents work? Let's build our intuition about why a^(-b) = 1/(a^b) and how this definition keeps exponent rules consistent. Continue the pattern of decreasing exponents by dividing by 'a', and see how it extends to zero and negative powers. While we're at it, we'll see why a^0 =1. Created by Sal Khan.

## Want to join the conversation?

• What is understanding exponents useful for? can it ever be used in daily life?
• You can use them for taxes, material management, and funds.
• Why do we even use exponents; when will we ever even use them in life?
• Here are some real life applications of exponents.

1. Calculations of areas (including surface areas) and volumes of objects
2. Calculations of distances in situations involving right triangles (Pythagorean Theorem)
3. Calculations involving loans or savings accounts, when interest is compounded
4. Calculations of probabilities of compound events
5. Calculations pertaining to motions of objects (for example, the height of an object thrown in the air as a function of time)
6. Expressing very small or very large measurements in science (for example, using scientific notation to express the mass of an electron or the mass of a planet)
7. Geometric Brownian motion model of the behavior of stock prices
8. Calculations in statistics, such as variance and standard deviation
9. Population growth or decay models

Have a blessed, wonderful day!
• I wonder if Sal ever looks at the comments
• Could somebody explain going backwards with exponents? It's a little bit difficult to understand.
• Think of this pattern:
2^3=8
2^2=4
2^1=2
2^0=1
2^-1=1/2
2^-2=1/4

See how we have a pattern of dividing by two every time? So going down in exponents equates to dividing instead of multiplying!
• does anybody know what a and b are??
• a and b are variables that stand for any number.
• what is 0 to the 0th power
• it is undefined, since x^y as a function of 2 variables is not continuous at the origin
• I love knowing the WHY of these conventions, makes everything in math feel so much more solid, symmetrical and wonderful
• what is 0 to the 0th power
• Interesting question.

Thing is, unlike most things in Math which are defined to be a certain thing, 0^0 isn't. It can be equal to 0, 1 or can be undefined based on context.

That's really all I can say, as explanations for each case will delve into concepts which you probably haven't learnt yet. But, once you learn concepts like "combinations", "limits" and "power series", the different definitions of 0^0 will become clear.
• Sal, you are a gift to the mankind. May you live to be 200 and be in good health and always be blessed with success and prosperity.
• Amazing video. But I still have a question that is not leaving my mind, why is 0^0 equal to 1? Ain't 0^1 equal to 0... I'm baffled.
• Because 0^0 is either defined as 1 or left undefined.

0^1 is equal 0 x 0 or 0 times itself, if you have nothing and multiply it by nothing, you will still have nothing!

Happy Learning!

## Video transcript

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.