 Okay, today in this video I'm going to talk about properties of exponents. I believe these are the last two properties I'll talk about for exponents. We've got to talk about negative exponents and we've got to talk about zero exponents. In my very first video about properties of exponents, it did come up that we did have a negative exponent when dealing with one of the properties. I didn't quite explain everything about that. So in this video, I'm going to explain more about negative exponents. And then also, what happens if you have a zero for an exponent? Explain a little bit behind that also. Okay, so in this video, I will first talk about the algebra behind this. So I'm going to use variables to kind of explain these properties. And then with those, I'm going to replace all those variables with numbers, so we can get a better understanding of these two properties. Okay, so negative exponents, a negative exponent would be just a quick example. If I had, I think in my last video, this is what it turned out to be. If I had three to the negative third power, something to that effect. What happens when I have a negative exponent? Well, what I'm going to do real quick is I'm going to try to remember back to that previous video, and how do you get a negative exponent? Okay, so actually, I'm going to come go, I'm just going to do a number example real quick before I start writing down some algebra. So what if you have a problem where you have three to the second power divided by three to the fifth power? Okay, now when you look at these two numbers, three to the second, that's actually just a nine. Three to the second is just a nine. And then three to the fifth power, well that's, let's say, three times three times three times three times three. So this here is a nine, this here is a nine, that here is an 81. 81 times three would be 243. It's good to know a little bit of mental math when doing this. Okay, so basically what we have here is we have three to the second power, which is nine, and three to the fifth power, which is 243. That's just a small fraction. That's all that really is. It's just a small fraction. The number on top is small. The number on bottom is pretty big. So that right there is a number less than one. All right, now the thing is, now that we know that that's a pretty small number, what we can do, actually that number reduces here. I won't quite reduce it yet. But anyway, what we can do is instead of evaluating the number, instead of doing all that mental math that I just did, what we can actually do is we can simplify that beforehand. So I'm going to go a step backwards, I'm going to go a step backwards. And what I'm going to do is I'm going to use my quotient rule if I'm dividing, if I'm dividing like bases, three and three of the base, if I'm dividing like bases, all I need to do is subtract the exponents. So in this case, I'm taking three to the two minus five. The two and the five, you take two minus five, we just do the top number minus the bottom number. Okay, which becomes three to the negative third. So again, I was trying to explain where that three to the negative third came from. There it is. Now, what does that mean? What does that really mean when I have a negative exponent? Well, negative exponent simply just means a fraction. That's all it really means. Negative exponent is not a negative number. That's a really big mistake. That's a really common mistake that a lot of students will make when they first learn about negative exponents. A negative exponent is a fraction. That's all it is. It is not a negative number. So in this case, three to the negative third is actually one over three to the third. If I have a negative exponent, I need to make it into a fraction. That's one way to explain it. I need to take this number, change its position. Right now it's on the top of a fraction, you can think. You can think of a fraction bar here with a one underneath it if that helps you. I need to take this number and I need to take it to the bottom of the fraction and make the bottom of this fraction a little bit bigger. So if I have a negative exponent, change its position, make it into a fraction. So I have one over three to the third, which is actually going to be one over, what is that? Three times three is nine times three is 27. Okay now earlier, I actually wrote out this fraction beforehand. It was nine over 243 and I mentioned that fraction reduces. So actually let's reduce that fraction and see what happens. Both of these numbers are divisible by three, nine is obviously divisible by three, 243, three goes into 24 and three goes into three. So I know it divides evenly. So three goes into nine three times, three goes into 24, 81 times, notice the three and the 81, some similar coincidental numbers. So actually I know nine, three will go into 81 and three will go into three. So three goes into 81, three goes into eight twice, remainder two, 21, three goes into 21 seven times. So that nine over 243 reduces to one over 27. So notice what we have here, one over 27, one over 27. So either way you look at it, you're going to get the same result. But notice the different ways to do this, you can either just take the bases, take the exponents, subtract them and make it into a fraction, which is actually an easier process than having to do nine over 243 and do all the mental math of reducing that fraction. So again, either process will work, it just kind of depends on what you like best. Now the thing is, once we start introducing variables into this, we can't really use fractions, we can't be reducing some of them, forced to use this negative exponent property. Okay, so that's where negative exponents come from. So let's actually get into the algebra behind this. So if I have some, a base to a negative power, if I have a base to some negative power, that's going to be basically, as I said before, just take that and make it into a fraction, this is going to be one over, make it into a fraction, one over b to the negative x. Oh excuse me, to the positive x. Once I make this b, once I make my base into a fraction, the exponent goes from negative to positive. So notice the change here, base to a negative exponent, you flip this base and you have a positive exponent. So let's actually do this with numbers similar to that last example that I did. Actually I'm going to do a couple of examples here. So if I have a regular number, maybe eight to the negative second, all I'm going to do with that is I'm just going to flip it. It's just going to be one over eight to the second. Now notice that the variable, or excuse me, the exponent went from negative two to positive two. So when I flip it, when I switch it, it becomes a positive exponent and that's just going to be, if I evaluate that, that's just going to be one over 64. I didn't evaluate the last ones, but that one was easy enough to do. Another example I want to do with numbers, what if you have a fraction? What if you have two thirds to the negative second, something like that? Okay, now what you're going to do actually is you're just going to, and now this holds true more here with the algebra, you're going to take your base and you're going to flip it and you're going to take your exponent and make it positive. So take the base, flip it, and now your exponent is positive. So that actually becomes, if we can see it on the edge of the board here, three to the second over two to the second. Okay, I won't evaluate that one, actually it would be nine over four, simple enough. We're just doing the basics behind this. Okay, so that's negative exponent. Again, sometimes that can be real confusing. Sometimes the algebra behind that can be confusing. Just look at the number of portions over here, I like these two examples, eight to the negative second becomes one over eight to the second. Now this basically just brings it to the bottom of the fraction, change its position I like to say. All right, so that's negative exponent, zero exponent, so what if you had this scenario, so I'm not going to start with the algebra, start with a quick example. What if I had three to the fifth power divided by three to the fifth power? What would happen? Okay, if I had three to the fifth power over three to the fifth power, if I use my same rules with the quotient property, I need to subtract these numbers, these exponents. So this would be three to the fifth minus five, which is three to the zero power, which is oh my goodness, what is that, okay? So when this situation comes up, if I ever have zero for an exponent, a lot of students will think that's crazy, what do I have zero there for? It's actually not that complicated, instead of looking at the zeros here, let's look backwards. Let's go back to the very beginning of the problem. Three to the fifth over three to the fifth. Now the thing is, this is actually very similar if I had six over six. Six over six reduces to one. So the thing is, this fraction is simply just the same thing on top and the same thing on bottom, same thing on top, same thing on bottom. So actually this fraction just reduces to one. So actually if I have a number to the power of zero, it's not going to be a question mark what the answer is, it's actually just simply going to be one. Any number to the zero power is just going to be one. So that's kind of an example to lead us into this. Now let's actually do this with the algebra and some number examples. Okay, so if I ever have a base to the zero power, it's always going to be equal to one. Now this number, this base is anything. It could be five, it could be ten, it could be a million, it could be a negative number, it could be just about anything. Any number to the zero power is going to be one. Any number to the zero power. So it really doesn't matter. Let's use a negative number, why not? Negative ten to the zero power is going to be one. Any number to the zero power is going to be one. And again, it could be a fraction, it could be a decimal, it could be just about anything. But there's just a number example to better explain that. Okay, that was a little bit of a long-winded video, but again, I had to do a little bit of a couple of examples to lead us into what a negative exponent and what a zero exponent is before writing down all these examples. So there we are, the properties of exponents. That is the negative exponent property and the zero exponent property. Just remember for negative exponents, take your base and flip it. That means you have your exponent, your negative exponent becomes positive. Look at these number examples to help you out with that. Zero exponents, any number to the zero power is always going to be equal to one.