 What we've done so far is go from the real number set, we've got our natural numbers, whole numbers, integers, rational and irrational numbers. What we're going to do is move over to the exponents. So what we're going to do is we're going to take everything that we learned in the real number set and move over to the exponents. We've already talked about the subsections in the exponents. Now what we're going to do is bring over addition, subtraction, multiplication and division. Now see how the operations in the rational number set apply to the exponents. Now obviously, well actually not obviously, sometimes it's easier to do multiplication and division, sometimes it's easier to do subtraction and addition. Right now what we're going to talk about, we're going to deal with subtraction and addition and basically same thing, we're going to group those things together and we're going to group multiplication and division together. Multiplication and division, the way it works is the base has to be the same. That's all you need. The exponent can be anything, anything. So as long as the base is the same, if you're multiplying or dividing, let's call this division or multiplication, as long as the base is the same, the exponents can be anything. So let's call anything, this is the paint box, and that's a blue box. They don't have to be identical, unlike adding and subtracting. So the way it works is if you had a squared times a cubed, the rule for multiplying and dividing exponents is, if you're multiplying, you add the exponents, if you're dividing them, you subtract the exponents. So this is multiplication. The bases are identical, so you're okay there. That means the exponents can add, so 2 plus 3. So this becomes A2 plus 3, which is 5. So this just becomes 8 power of 5, and that's your answer. So when it comes to multiplication and division, the bases have to be the same. The exponents can be anything. Let's do a few random. So if you're multiplying two things that the bases are the same, the exponents are different. Well, the exponents can be the same or different. You just add the exponents. If you're dividing two things that the bases are the same, you subtract the exponents. So let's do an example quickly. So right now we've got 120 A to the negative 2 B cubed divided by 40 A to the negative 5 B to the 5. Now what we do with the exponents, their bases are the same. The numbers, the coefficients are in front. I think the numbers in front are just called coefficients. The numbers in front, they just deal together exactly the way they did in the real number set, right? So we're talking about the first way, just adding or subtracting, multiplying, dividing numbers. So 120 divided by 40 is going to be 30. A to the negative 2 divided by A to the negative 5. What you do with division, you subtract the exponents. So it becomes negative 2 minus negative 5. And this becomes A negative 2 minus negative 5. And you do the same thing with the B, because the bases are the same. So it becomes 3 minus 5. So B, 5. So we talked about this before, and negative and negative becomes positive. So this becomes plus. So this becomes 30. A to the power of negative 2 plus 5 is 3. B to the power of 3 minus 5 is negative 2. Right? And we talked about what negative powers mean when we're laying out all the transition or moving on from the rational number set to the exponents. And we said negative numbers. All it does is just kicks it down, right? If it's on the top, it goes to the bottom. Bottom, it goes to the top. So the way this works is, now this negative 2 only applies to the B. It doesn't apply to the A more than 30, because these guys are in brackets to the negative, right? So the way it works is, you can pull down here and pull down here. So this becomes 30. A cubed that stays on top. Over. That's our answer, okay? So that whole thing just reduces down to 30 A cubed B squared. You have something like this. A to the power of a half times A to the power of 5 over 3. What you've got to do, you've got to add the exponents. So this becomes A to the power of a half plus 5 over 3. And the way you add these is, well, you go A. You've got to find a common denominator. A common denominator of 2, 2 and 3 is 6. You multiply this by 3. So you multiply that by 3. So it becomes 3 plus. You multiply that by 2. 3 by 2 gives us 6. 2. So it becomes 10. So this thing multiplied together is going to be 10 to the power of 13 over 6. All you do is you add the exponents. And it doesn't make a difference how many different symbols, variables you have multiplying together and dividing together. So for example, you have, so we have 2 to the power of 5 A squared B to the negative 2 times 2 to the negative 3 A to the negative 5 B to the 5. And the way it works is all the bases, they're the same. Even if you had, let's say you had a w. We don't have a w here, but that's okay. Because all you do, you put w to the power of 0, but that's going a little too far, too far. So all you do is you deal with the exponents. So 2 to the power of 5 times 3 to the power of negative 3. All you do is just add the exponents. 5 plus negative 3 is going to be 2. So it's 2 squared. A squared times A to the negative 5. 5 plus negative 5 is going to be A to the negative 3. Negative 2 plus 5 is going to be 3, right? So it's going to be B to the power of 3. And your w is just, your w is just so... Now what's the final answer for this? Well, 2 squared is just going to be 4. 2 times 2, you're floating itself. A to the negative 3. We talked about this in the layout of the exponents when we're transferring over the information from the rational number set for the exponents or laying down the properties of exponents. And negative power means you flip it, right? And the only thing that flips here is the A because that's the only one to the negative power. Now if it was negative 3 to this whole thing, this whole thing could have been negative power, the whole thing would have flipped. But the negative 3 right now only applies to the A. So this becomes divided by A to the power of 3. You got B cubed and w on top. And that's your final answer. So when you're multiplying, dividing, bases with exponents that are the same, you just add or subtract the exponents depending on if you're multiplying or dividing. Now let's do a little bit more complicated exponent to a fraction. Let's assume you had... The way this works is this is just multiplication. So anything that has the same base, you can combine their exponents. So this is multiplication. So it becomes A squared times A negative 3 over 2. So you're just adding those guys. And for the B, you're just adding those guys because it's still multiplication, right? So this would be A 2 plus negative 3 over 2 plus. Now, the way you do this, you've got to do your common denominator. The common denominator, that's just over 1. The common denominator here is 2. So this would be A to the power of the common denominator is 2. You multiply the 1 by 2. So that becomes 4. And that stays the same. Minus 3. The B becomes the common denominator between 2 and 7 is 14. Multiply this by 7 so that becomes 7. Let's multiply that by 2 so that becomes... So we've got this. This. 4 minus 3 is 1. So this becomes A to the power of a half. Right? And for the B it's 7 plus 10 is 17. So it's B to the power of 17 over 14. 17. Does that make sense? Hopefully.