Multiplying monomials | Polynomial arithmetic | Algebra 2 | Khan Academy

Let's say that we wanted to multiply 5x squared, and I'll do this in purple: 3x to the fifth. What would this equal? Pause this video and see if you can reason through that a little bit.

All right, now let's work through this together. Really, all we're going to do is use properties of multiplication and use properties of exponents to essentially rewrite this expression. We can just view this as 5 times x squared times 3 times x to the 5th, or we could multiply our 5 and 3 first.

So you could view this as 5 times 3 times x squared times x to the fifth. Now, what is 5 times 3? I think you know that that is 15. Now, what is x squared times x to the fifth? Some of you might recognize that exponent properties would come into play here. If I'm multiplying two things like this, we have the same base and different exponents.

This is going to be equal to x to the, and we add these two exponents: x to the 2 plus 5 power, or x to the seventh power. If what I just did seems counterintuitive to you, I'll just remind you: what is x squared? x squared is x times x, and what is x to the fifth? That is x times x times x times x times x.

If you multiply them all together, what do you get? Well, you got seven x's, and you're multiplying them all together. That is x to the seventh. And so, there you have it: 5x squared times 3x to the fifth is 15x to the seventh power.

So the key is, look at these coefficients. Look at these numbers: the five and the three. Multiply those, and then for any variable you have, if you have x here, so you have a common base, then you can add those exponents. What we just did is known as multiplying monomials, which sounds very fancy.

But this is a monomial. Monomial! And in the future, we'll do multiplying things like polynomials, where we have multiple of these things added together. But that's all it is: multiplying monomials.

Let's do one more example, and let's use a different variable this time just to get some variety in there. Let's say we want to multiply the monomial 3t to the seventh power times another monomial: negative 4t. Pause this video and see if you can work through that.

All right, so I'm going to do this one a little bit faster. I'm going to look at the 3 and the negative 4, and I'm going to multiply those first, and I'm going to get a negative 12.

Then, if I were to want to multiply the t to the seventh times t, once again, they're both the variable t, which is our base. So that's going to be t to the seventh times t to the first power. That's what t is. That's going to be t to the seven plus one power, or t to the eighth.

But there you go! We are done again. We've just multiplied another set of monomials.