 What we've done so far is go from the real number set, we've got our natural numbers, pole numbers, incongrues, rational and irrational numbers. What we're going to do is move over to the exponents, right? So what we're going to do is we're going to learn, 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 is basically the same thing, we're going to group those things together and we're going to group multiplication and division together. So let's go over and talk about adding things that are to the exponent and see where that goes, how that goes. When you're adding stuff that are to the exponents, addition and subtraction basically are sort of a universal rule where you can only add and subtract like things. So a box can only add or subtract from a box. A box plus or subtract the triangle is going to be a box plus or add, adding and subtracting a triangle. So you can only add or subtract like terms. So if you have something, let's say, if you go down here, if you have a to the power of two plus a to the power of three, you can't add those because they're not completely identical. When you're adding and subtracting, the base has to be the same and the exponent has to be the same. So it's irrelevant if both of these have the same exponent. If their bases are not the same, you can't add them. Over here we have two things that have the same base but different exponents. And you can't add these things. So that would be equivalent to wanting to add a square and a square but their exponents are different. So this would be a square and this would be a triangle. You can't add those. So keep this in mind. We're going to do a few examples. Very straightforward. Plus a. That's okay. A plus a. Two. If you have two, let's put a square there. Two a squared plus three a squared. Now they're exactly the same thing. So it doesn't make a difference what's in the box as long as the boxes are identical you can add them. So two boxes plus three boxes is five boxes. And what's the box we're talking about is a squared. So a squared. And this continues on as long as they're completely identical you can add them. So when you're combining things, both the base and the exponent have to be identical. So if I gave you something like this a squared times the e plus a squared you can add these because of multiplication. These two things are multiplied together. So this supersedes this, the addition part anyway. So the answer to this would just be a squared. a squared b plus a squared. So this would just be a squared b plus a squared. It doesn't reduce anything. So when you're adding or subtracting, they have to be identical. It doesn't make a difference in what they are. It could be extremely complicated. For example, if you had w over z let's make two of these guys plus w then the answer to this would just be two plus five. Whatever this thing is because they're identical. So it doesn't make a difference in what it is that you're adding as long as there's no difference between them. For example, if this was a power of two and that's a power of one, you couldn't add them because that wouldn't work out. Right now the way it stands, the answer to this would be seven w over z. If there's any difference between them, you cannot add a subtractive. So for example, if you had those two things you can add because the term here is identical. They're identical to each other. So the answer to this would just be three minus eight which is negative five. So this would be negative five w squared square root of z over q. I hope that's pretty clear where if there's any differences between these two guys, you can't add or subtract them. So for example, if this was w squared z square root of z q squared, if this was a q squared then you couldn't add those. That wouldn't be the answer. This is super important because once you get into radicals, with radicals you can only add and subtract like terms as well. So for example, if you had powers just like this, if you had three square root of five minus eight square root of five, then the answer to this is negative five square root of five because it's just three minus eight, right? If there's any difference between these radicals you can't add or subtract them. So for example, if you had three square root of five minus, you can't do anything with this. The answer to this would just be itself. Mix radicals, adding subtracting mixed radicals. So you could have something like let's just stick with two square root of eight. Two square root of eight. And then let's go minus five square root of thirty. Now two square root of eight. We just did this, but it would be two, two, two. Paramount, they come out of singles. There's a two waiting for them. So it multiplies the two. It becomes four square root of two. Thirty-two is five twos multiplied together. It's four times eight, four breaks down to two. And eight breaks down to eight, two, two, two. So it becomes two, two, two, two, two, two. Square root means bring out those two. Those two twos is a single two. Bring out those two twos is a single two. So two times two is four. It comes out as four, because those two come out as a two. Those two come out as a two. What's waiting for them is a five. Times five is twenty. Square root of two. Now, because these things are exactly the same, you can combine your like terms. So four minus twenty, four root two minus twenty, is just four minus twenty. Four of this minus twenty. That's the same thing. So it's just become negative sixteen. And that just means it's a negative number. This is not a negative in the exponent where you flip the whole thing. It's a negative number. Don't get confused. I'm going to say it again. Don't get confused between what's in the base and what's in the exponent. This is just a negative number. And that would be the answer to that. Because when you get radicals like that, you have to break them down so they're identical radicals for you to be able to combine them. Otherwise, you can't combine them.