Video Transcript: Negative Exponents

Negative exponents 

We already know that 2 to the fourth power can be viewed as starting with a 1 and then  multiplying it by 2 four times. So let me do that. So times 2, times 2, times 2, times 2. And that  will give us, let's see, 2 times 2 is 4, 8, 16. So that will give us 16. Now I will ask you a more  interesting question. What do you think 2 to the negative 4 power is? And I encourage you to  pause the video and think about that. Well, you might be tempted to say, oh maybe it's  negative 16 or something like that, but remember what the exponent operation is trying to do.  One way of viewing it is this is telling us how many times are we going to multiply 2 times  negative 1? But here we're going to multiply negative 4 times. Well, what does negative  traditionally mean? Negative traditionally means the opposite. So here this is how many times  you're going to multiply. Maybe when we make it negative this says, how many times are we  going to, starting with the 1, how many times are we going to divide by 2? So let's think about  that a little that. So this could be viewed as 1 times, and we're going to divide by 2 four times.  Well, dividing by 2 is the same thing as multiplying by 1/2. So we could say that this is 1 times  1/2, times-- let me just do it in one color. So 1 times 1/2, times 1/2, times 1/2, times 1/2. Notice  multiplying by 1/2 four times is the exact same thing as dividing by 2 four times. And in this  situation this would get you, well 1/2, well 1 times 1/2 half is just 1/2, times 1/2 is 1/4, times 1/2 is 1/8, times 1/2 is 1/16. And so you probably see the relationship here. If you're-- this is essentially you're starting with the 1 and you're dividing by 2 four times. You could also say that 2-- I'm  going to do the same colors-- 2 to the negative 4 is the same thing as 1/2 to the fourth power.  Let me color code it nicely so you realize what the negative is doing. So this negative right over  here-- let me do that in a better color, I'll do it in magenta, something that jumps out. So this  negative right over here, this is what's causing us to go one over. So 2 to the negative 4 is the  same thing, based on the way we've defined it just up right here, as reciprocal of 2 to the  fourth, or 1 over 2 to the fourth. And so you could view this as being 1/2 times 2 times 2 times 2, if you just view 2 to the fourth as taking four 2's and multiplying them. Or if you use this idea  right over here, you could view it as starting with a 1 and multiplying it by 2 four times. Either  way, you are going to get 1/16. So let's do a few more examples of this just so that we make  sure things are clear to us. So let's try 3 to the negative third power. So remember, whenever  you see that negative, what my brain always does is say I need to take the reciprocal here. So  this is going to be equal to, I'm going to highlight the negative again, this is going to be 1 over 3  to the third power. Which would be equal to 1/3 times 3 times 3, or 1 times 3 times 3 times 3, is  going to be 27. So this is going to be 1/27. Let's try another example, I'll do two or three more.  So let's take a negative number to a negative exponent, just to see if we can confuse ourselves.  So let's take the number negative 4, and let's take it-- I don't want my numbers to get too big  too fast. So let's just take negative 2 and let's take it to the negative 3 power. I'll make my  negatives in magenta, negative 3 power. So at first this might be daunting, do the negatives  cancel? And that will just be the remnants in your brain that are trying to think of multiplying  negatives. Do not apply that here. Remember, you see a negative exponent, that just means  the reciprocal of the positive exponent. So 1 over negative 2 to the third power, to the positive  third power. And this is equal to 1 over negative 2 times negative 2 times negative 2. Or you  could view it as 1 times negative 2 times negative 2 times negative 2, which is going to give you 1 over negative 8 or negative 1/8. Let me scroll over a little bit, I don't want to have to start 

squishing things. So this is equal to negative 1/8. Let's do one more example, just in an attempt  to confuse ourselves. Let's take 5/8 and raise this to the negative 2 power. So once again, this  negative, oh I got at a fraction is a negative here. Remember this just means 1 over 5/8 to the  second power. So this is just going to be the same thing as 1 over 5/8 squared, which is going to  be the same thing-- so this is going to be equal to-- I'm trying to color code it, 1 over 5/8 times  5/8, which is 25/64. 1 over 25/64 is just going to be 64/25. So another way to think about it is,  you're going to take the reciprocal of this and raise it to the positive exponent. So another way  you could have thought about this is 5/8 to the negative 2 power. Let me just take the  reciprocal of this, 8/5 and raise it to the positive 2 power. So all of these statements are  equivalent. And that would have applied even when you're dealing with non-fractions as your  base right over here. So 2, you could say well this is going to be the same thing. 2 to the  negative 4 is going to be the same thing as taking my reciprocal. So this is going to be the same thing as taking the reciprocal of 2, which is 1/2 and raising it to the positive 4 power. 

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.