Multiplying monomials | Algebra I | Khan Academy

All right, in this video we're going to be multiplying monomials together. Let me give you an example of a monomial: 4 x squared. That's a monomial. Now, why? Well, mono means one, which refers to the number of terms. So this 4x squared, this is all one term.

So what? So we're going to be working with things like that. What won't we be working with? Well, what about 4x squared plus 5x? How many terms are there? 4x squared is the first term; 5x is the second term. So this is not a monomial. This is actually called a binomial because bi means two, like your bicycle's got two wheels, for example.

So not yet, go on to the future videos if you're ready for binomials, but we're just going to be working with multiplying monomials together. So can we grab an example to look at? By the end of this video, it should be very easy for you to multiply this monomial, 5x squared, by this monomial, and I'm actually going to just give you the answer right here. And then I'm going to slowly walk you through some other questions that will lead us to why.

But the answer to this is 20 x to the eighth. 20 x to the eighth. Take a look at that; see if you can notice a pattern. What do we do with the 5 and the 4 to get the 20? What do we do with the 2 and the 6 to get the 8? That's getting a little ahead of ourselves, though. Before we can dive in there, let's remember some of the exponent properties, a very specific exponent property that you should have seen before.

So if we look at 5 squared times 5 to the fourth power, what's that going to equal? Well, if you remember your exponent property—and we'll do a quick reminder here—I always add my exponent. So 5 squared times 5 to the fourth power is equal to 5 to the sixth power. What about three to the fourth power times three to the fifth power? Well, again, I always add my exponents. Four plus five is three to the ninth power, and my base of three stays the same. Great!

So if you remember that, now we're ready to really start multiplying monomials that are new to you, and the new thing here is that we are going to have variables involved. So let's start by... let's take a look at two monomials here. The first monomial is 4x, and the second one is just x. And the 4, I don't have another number to multiply by; I've just got the 4. And can I simplify x times x? Well, that's equal to x squared.

Remember, if I just have a variable and there's no exponent there, it's equivalent to having a 1. So x to the first power times x to the first power, I add my exponents, like we just talked about, and one plus one is equal to two. Great! So let's move on to another one here: five four t times 3t. Well, 4 times 3 is going to be equal to 12.

So I've combined my coefficients, and then t times t, again think of a 1 being there, is going to be t squared. So the answer here is 12 t squared. So let's keep going, and once you get into the rhythm of these, they become pretty... all right!

So what if I had 4p to the fifth power times, let's say, 5p to the third power? What would that equal? You're going to notice a pattern here that we've been picking up on, which is that I'm always going to multiply my coefficients. So 4 times 5 is going to equal 20, and I'm always going to add my exponents.

So p to the fifth and p to the third is p to the eighth power. So I multiply four and five to get twenty; I add five and three to get eight. And if you really want to see why that is, let's really dive in here and let's break down this first term. Let's break down four p to the fifth. I can write that out as four times p times p times p times p times p. That's five of them—it's four and five p's.

And then that second term I can write as times 5 times p times p times p. What I'm going to do is I'm going to group my numbers because I can work with numbers together. So let's put 4 times 5 at the very front, and then it just becomes a matter of how many p's do I have. We'll put all of those together as well.

So I had five p's, so there's the first five, and then I had three more. And we can simplify this crazy-looking expression by just multiplying by 4 my 5 to be my 20 and then writing this with an exponent. That's the beauty of exponents—that's why we have them. Is we can write a crazy expression like that as p to the eighth, and you'll notice that this is, of course, what we got the first time.

So great! What about five y to the sixth times negative 3 y to the eighth power? Again, multiply the coefficients, add the exponents, and I've got a simplified expression. Let's get really crazy here; let's have a little fun. So we've noticed the pattern.

Let's have a little fun. You're saying, "I can do more!" Negative nine x to the fifth power times negative three. Use parentheses there; you always, when you have a negative in front, you want to use parentheses. Let's do x to the 107th power.

If I would have showed you this before this video, you would have said, "Oh my goodness! There's nothing I can do! I'm boxed! There's no way out!" But now you know that it's as simple as follow the rules. We're going to multiply the coefficients. Negative 9 times negative 3 is 27. Two negatives is a positive, and nine times three is 27.

I'm going to add my powers: five plus 107 is a hundred and... Not two! That was almost a mistake I made there! Let's get rid of that. Give me a second chance here! It's all about second chances! Five plus 107 is 112.

And so this crazy expression, which is two monomials—here's the first, here's the second—when we multiply and simplify, we get another monomial, which is 27 x to the 112. I'm going to leave you on a cliffhanger here, which I'm going to show you a problem.

What variable should we use? You notice I've been trying to vary the variables up to show you that it just doesn't matter. That's an ugly five; let's get rid of that! Give me a second chance on that one too! So let's look at 5 x to the third power times 4 x to the sixth power, and I'm going to show you a wrong answer.

I had a student that I asked to do this, and here's the wrong answer that they gave me. They told me 9 x to the 18th power. That's terribly wrong. What did they do wrong? What did they do wrong? I want you to think to yourself, what have we been talking about? What did they do with the 5 and the 4 to get the nine? What should they have done?

What did they do with the three and the six to get the 18? And what should they have done? That's multiplying monomials by monomials.