Welcome back, everyone. We saw how to simplify expressions by combining like terms. So, for example, in this expression, we could combine x2+2x2 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. What I'm going to do in this video is show you by using all these rules we're going to talk about that this expression actually just simplifies down to something like x∙y. 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 1 to the 4th power, 1 to any power, and I wanted to know what that evaluates to. Well, that's just one to the 4th power which 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. 1 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 three 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 2 times negative two times negative two. So 2 times two times two is just 8. But what happens to the negative sign?
Well, this gets canceled out with this one. But what about this one? This third negative sign doesn't have another one to cancel it out so the negative sign actually just gets kept there. So this is negative 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 multiplication and division. Let's see what happens when you have something like 42∙41. We'll just write this out. 42 is 4 times 4, then we multiply by another factor of 4. It's all multiplication. So it's basically like I just have 3 fours multiplied together. But the easiest way to represent this 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, the 2 and the 1, we 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 same, but one is tilted. So it's an easy, silly way to remember this. But that actually turns out to be a 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+y70, and you can actually really simpl