 In this video, I'm going to talk about a couple more properties of exponents. These are just two more. There are a couple more that I still need to do videos on, but I'm just going to do two for this video. First thing I'm going to start with is, I'm going to start with the algebra behind this, so I'm going to use variables to explain these, and then I'm going to put numbers in for those variables to better understand it. Okay, the two properties I'm going to talk about today are the power of a product and the power of a quotient. First that we have product and quotient, so we're going to have to deal with multiplication and we're going to have to deal with division for these two separate properties. Okay, power of a product basically states that, actually, you know what, let me write down an example first and then I'll explain it. It's a little bit easier to see with an example. Okay, so I've been using a couple of different variables. So if I have A times B to some power, we'll call it X. Okay, so what we're doing here is we're taking the power of a product. This power right here, this X, where this power is being applied to a product. A times B, that's a product. Okay, so we're taking a power of a product. That's where we get this name from, power of a product. Okay, so power of a product simply just states that you can actually take the individual numbers inside of the parentheses and kind of distribute the exponent evenly between those two. This is what it kind of looks like. So I can also take, if I have A times B, this quantity to the X, I can actually take this X individually to each one of these on the inside. Kind of like distributive property, not exactly the same thing, but very similar to it. So I, in this case, I would have A to the X power times B to the X power. Notice I took this X and applied it to the A. I took the X and applied it to the B. So I have A to the X times B to the X. All right, so that's with the variables. Now let's do that with numbers. Okay, so a couple of different numbers here. For A, I'll choose, so I'll just keep these numbers simple. A, I'll choose two. B, I'll choose three. And X, I will choose five. Okay, just to keep the numbers real simple, we don't have to complicate things out very much. Now, one of the first things that you notice when you see this property is that, well, I see two times three inside of the parentheses. Can I just do that first? Yes, absolutely, you can do that first. You can take two times three, make that six, and then this would be six to the fifth power. And you wouldn't have to distribute this X. Okay, but what we're doing is we're just going kind of a different method of simplifying this, different method of type of solving that we do with this. This isn't usually used with numbers, this method is usually used when you have variables, a lot more variables than what we have here. But anyway, so to kind of do this with numbers, do this example with numbers, if I have a product that's being, a power that's being applied to a product, well, I can take this power and distribute it to the two terms inside. So this is going to be equal to two to the fifth power times three to the fifth power. Okay, now from there, we can evaluate what that is. Two to the fifth, plug that into a calculator, we know what that is. Three to the fifth, plug that into a calculator, we know what that is. Some of you might be able to do that off the top of your head. Am I going to evaluate that anymore? I'm just going to leave that where it's at, but you can continue to simplify that now. Okay, so that is the power of a product. Next, I'm going to do the power of a quotient. So as you well can imagine, if I'm doing the power of a product, this one is going to be the power of a quotient. So actually, I'm going to divide inside of my parentheses and the exponent is going to be on the outside. Power of a quotient, power of a quotient. Okay, so that's again where we get the labeling for these. Okay, so power of a quotient, very similar to the power of a product. I'm going to take that exponent and kind of apply it to each one of my terms inside. I'm going to take that x and apply it to the a. I'm going to take that x and apply it to the b also. All right, so it looks a little bit like this. Again, very similar to what we did up here, but again, it's a different rule. Instead of multiplying, we're dividing here. So this is going to be a to the x power over b to the x power. Again, very similar. You can almost think of this as distributive property, not really. It's not exactly distributive property. I kind of like to think of it that way though. Take this x and apply it to the a, take this x and apply it to the b. So take that power, apply it to the top of the fraction, take that exponent, apply it to the bottom of the fraction. Okay, so that's the algebra behind it. Now let's use numbers. I'm going to use the same numbers I used last time. So in this case, a was 2. So I have 2 divided by 3 in parentheses. And that's being taken to the fifth power. Okay, now last time I kind of explained, with the power of a product, you can multiply these numbers. Now down here, 2 divided by 3, that doesn't divide evenly. So you don't really want to divide that first. But sometimes there will be numbers in here. You can divide that first before you start applying the exponent. Again, these rules aren't going to work every time. Sometimes we'll use them. Sometimes we won't. It just kind of depends on what the problem asks us to do. Okay, anyway, 2 thirds to the fifth power. So I'm going to take that exponent and apply it to the top of my fraction, take the exponent and apply it to the bottom of my fraction. So this is going to be 2 to the fifth power on top, 3 to the fifth power on bottom. And again, just like last time, I can continue to evaluate that. I can actually figure out what 2 to the fifth is and keep going and simplify this. I can figure out what 3 to the fifth is and continue that. But just for this short video, I'm not going to continue. Those that don't have a lot of room with this. But those are the two properties of exponents. Power of a product, power of a product, and the power of a quotient. Okay, so those are the two, those are two of the properties of exponents. There are a few more, but those are just the two now for this video.