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CCSS Math: HSA.APR.A.1

Video transcript

- [Voiceover] Alright, in this video, we're gonna be multiplying monomials together. Let me give you an example of a monomial. 4x 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 we're gonna 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's 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 gonna 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 just gonna give you the answer right here. And then I'm gonna slowly walk you through some other questions that will lead us to why. But the answer to this is 20x to the eighth. 20x to the eighth. Take a look at that, see if you can notice a pattern. What did we do with the five and the four to get the 20? What did we do with the two and the six to get the eight? 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've seen before. If we look at five squared times five to the fourth power, what's that going to equal? Well, if you remember your exponent property, we'll do a quick reminder here, I always add my exponent. So five squared times five to the fourth power is equal to five 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 there is that we are going to have variables involved. So let's start, let's take a look at two monomials here. The first monomial is 4x, and the second one is just x. And the four, I don't have another number to multiply by, just got the four. 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 one, 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. If I have 4t times 3t. Well, four times three is gonna be equal to 12, so I've combined my coefficients. And then t times t, again, think of a one being there, is going to be t squared. So the answer here is 12t squared. So let's keep going, and once you get into the rhythm of these, they become pretty alright. So what if I had 4p to the fifth power times, let's say 5p to the third power. What would that equal? Well you're gonna notice a pattern here that we've been pickin' up on, which is that I'm always gonna multiply my coefficients, so four times five, 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 til we get 20, I add five and three to get eight. And if you really wanted to see why that is, let's really dive in here and let's break down this first term, let's break down 4p to the fifth. I can write that out as four times p, times p, times p, times p, times p, that's five of 'em. That's four and five p's. And then that second term I can write as times five times p, times p, times p. What I'm gonna do is I'm gonna group my numbers, cause I can work with numbers together, so let's put four times five 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 my four and my five to be my 20, and then writing this with an exponent, that's the beauty of exponents, that's why we have 'em, 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 5y to the sixth times negative 3y 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. Just saying, I can, I can do more. Negative 9x to the fifth power times negative three, use parentheses there, when you have a negative in front, you always wanna 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 nine times negative three is 27. Two negatives is a positive and nine times three is 27. I'm gonna add my powers. Five plus 107 is a hundred, ooh, not two, that was almost a mistake I made there. Let's get rid of that, give me a second chance here. Life'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 27x to the 112th. I'm gonna leave you on a cliffhanger here. Which, I'm gonna 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 with that one too. So let's look at 5x to the third power, times 4x to the sixth power. And I'm gonna show you a wrong answer. I had a student that asked to do this, and here's the wrong answer that they gave me. They told me 9x 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 five and the four 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.