Welcome back, everyone. We saw how to simplify expressions by combining like terms. So for example, in this expression, we could combine x2 and x2 into 5x2 and make the expression shorter. But combining like terms isn't always going to work. In this complicated expression over here, we can't combine anything because there are no pluses and minus signs. So it turns out that when this happens and we can't combine like terms, we're going to need some new rules to simplify expressions that have exponents in them. And what I'm going to do in this video is I'm going to show you by using all these rules we're going to talk about that this expression actually just simplifies down to something like xy. It's pretty cool. Let me just show you how it works. Feel free to use the page right before this that has a master table of all these rules so you don't have to fill this out multiple times, and you'll have all your notes in one place. Let's just go ahead and get started here. So let's say I had something like 14, 1 to any power, and I wanted to know what that evaluates to. Well, that just ends up being 1 times 1 times 1 times 1, and it doesn't matter how many times you multiply 1 by. The end result is always just 1, and that's the rule. One to any power always just equals 1. Alright. So that's a pretty straightforward one. It's called the base one rule. The names are the least important thing about the rule. It's just really important that you learn how they work. Let's go ahead and move on to the second one here, a negative to an even power. So let's say I had -32. That just means negative three and negative three. 3 times 3 just equals 9. What happens to the negative signs? Well, as long as you have a pair of negative signs, the negative sign always just gets canceled out. It doesn't matter if the exponent is 2 or 4 as long as it's any even number. So for example, -34 just looks like this, and we'll see that 3 multiplied by itself 4 times is 81, and what happens is the negative gets canceled with this one and this negative gets canceled with this one. So anytime you have a negative number to an even power, you basically just drop the negative sign or it just gets canceled out. That's the rule. Now let's see what happens when you have negatives raised to odd powers, something like -23. Well, let's write this out. This is negative two times negative two times negative two. So 2 times 2 times 2 is just 8. But what happens to the negative sign? Well, this gets canceled out with this one. What about this one? This third negative sign doesn't have another one to cancel it out the negative, so it actually just gets kept there. So this is -8. So this rule is the opposite. Whenever you have a negative to an odd power, you actually end up keeping the negative sign on the outside. So you keep the negative sign here. Alright. So pretty straightforward. Let's take a look at another couple of rules here. Now we're going to get into like multiplication and division. Let's see what happens when you have something like 42 times 41. We'll just write this out. 42 is 4 times 4, then we multiply by another factor of 4. Remember the dot and the x just mean the same thing. It's all multiplication. So it's basically like I just have 3 fours all multiplied together. But the easiest way to represent that is actually just 43. That's the simplest way I can do that. And so if you look at what happened here with these exponents, there's 2 and the 1, you basically just added them, and that's actually what the rule ends up being. Anytime you're multiplying numbers of the same base, you actually just add their exponents together. So when you multiply, you add. One way you can kind of remember this is that the multiplication symbol and the addition symbol, they kind of just look the like the same, but one is tilted. So it's an easy, silly way to remember this. But that actually turns out to be a really, really important rule and a shortcut because sometimes you're going to have expressions where you don't want to write out all the terms like y30 and y70, you can actually really simply figure this out. This actually just ends being y100. Alright? So pretty straightforward. Now that's called the product rule by the way. And now let's take a look at the last one where you're now dividing terms that have the same base. So it's not 4 times 4, it's 4 divided by 4. And we'll see here that this is just 4 times 4 times 4 divided by 1 factor of 4. And remember from, from fractions, we can always cancel out one of these things and we're just left with like a one that's out here. It's kinda like an invisible one. And the easiest way to represent this is just 4 times 4, but that's just 42. Alright. So here we actually ended up adding the exponents, but here to get the 2, we actually ended subtracting the 3 and then and the 1. And so that's the rule. Whenever you are dividing terms of the same base, you subtract their exponents. Alright? So when you divide, you subtract. And one way to remember this is that you're doing division, which kind of looks like a little minus sign, so division is subtraction. Now one tiny difference here is that when you added the exponents, the order doesn't matter because 2 +1 is the same thing as 1+2, but in subtraction, it does matter. You always have to subtract the top exponent from the bottom. So so always do top minus bottom. Alright? So that's really important. Don't mess that up. Alright, everyone. So that's it for the first couple of rules. We'll take a look at more later on. Let's get some practice with these rules over here. We're going to simplify these expressions by using the exponent rules. Let's take a look at the first one. We have -59 divided by -56. So in other words, we have the same base that's being divided with different exponents. That just means we're going to use the quotient rule and we're going to subtract the exponents. Alright? So in other words, we're going to take this and this is going to be -53, but remember this actually is now a negative number raised to an odd power. So we can use the negative to odd power rule, and we keep the negative sign on the outside. And then if we wanted to evaluate this as a single number, this would just be -125. Alright? So let's move on now to part b. In part b, now we're going to start mixing up numbers and variables. We have 2x4 and 7x2 all divided by x5. So we have multiplication and division. Let's just deal with the multiplication first on top. So what happens is all this stuff is multiplied. So in other words, the 2 and the 7 multiplied to 14, and then you have x4 x2. So in other words, you're multiplying numbers or terms of the same base, so that means we can actually add their exponents. So the 2 and the 7 become 14, and the x4 and x2 becomes x6, and then we just have x5 on the bottom. So in other words, what happens is this just becomes 14x6 divided by, x5, and now what happens is we have the same base on the top and the bottom. So So now we can use the quotient rule for this, and we subtract the exponents. In other words, this is just 6, 14x6 minus 5, and this just becomes 14 x, to the, you know,x to the one power. In other words, just 14 x. Alright. So that's it for the second one. Let's go take a look take a look at the third one. Here we have just multiplication of a bunch of these terms here. We can do the exact same thing that we did with the numerator in this term. Everything is multiplied. There's no division. So the 6 and the 4 basically just become 24. And now if you'll notice here, so that's what the 6 and the 4 become, you have an x3 and then an x2. So in other words, when you multiply those two things together, you get x3+2 using the product rule, and then we also have y2 y5. So when you drop those 2 down and use the product rule, you get y2+5. So this whole thing is just the product rule. You just sort of match up all of the the terms that are similar, and then you use the product rule for each one of them. So in other words, this just becomes 24, and this is x5 power. So x5 power, and then we have y7 power, and that's your final answer. Alright, folks. So that's it for this one. Thanks for watching, and watch I'll see you in the next video.