Common Core Algebra I.Unit 1.Lesson 7.Exponents as Repeated Multiplication.By eMathInstruction

Hello and welcome to another common core algebra one lesson by E math instruction. My name is Kirk weiler, and today we'll be doing unit one lesson 7, exponents is repeated multiplication. Before we begin, let me remind you that a worksheet that's used in this lesson can be found, as well as a homework set by clicking on the video's description. Don't forget at the top of each worksheet we have a QR code that can be scanned with a smartphone or a tablet to bring you right to this video. Let's get into it. One of the many things that we're going to have to deal with this year and in future years in algebra are exponents. And I know that you've seen exponents before. But they are a way of indicating repeated multiplication. All right? We're going to extend those to a huge degree in algebra, both in this course and announce for two. We're used to exponents being positive integers. One, two, three. And if they're positive integers, then it means that we're multiplying by some quantity that many times. All right? It's very important to understand this, right? Multiplication is repeated addition. Repeated multiplication though is exponentiation or exponents. So let's start off real simple. Exercise number one says write out each of the following exponents right out what each of the following exponents means is an extended product and find its value. Let's see how many of these we can do without our calculator. All right, so two to the fourth. What this means is two times two times two times two. Now the thing is, I'm going to actually use the associative property of multiplication to really look at the problem like this. All right, now you might say all that did was make it more confusing. But what I really mean is I'm going to do two times two, which is four. Then I'm going to do four times two, which is 8, and then I'm going to do 8 times two, which is 16. So two to the fourth is 16. A little bit easier to read in the second or three squared is three times three. That's simple enough. Just 9. Likewise, 5 to the third, or 5 cubed is 5 times 5 times 5. And again, we can think of this as 5 times 5, which is 25, and then 25 times 5 is a 125. All right. That's it. So right now, what exponents mean is repeated multiplication. All right? So we're going to get some practice just with that idea. I'm going to annoy you by having you write out lots of these extended products. But it's important because any time you see an exponent, I want you to feel confident that all it means is repeatedly multiplying by whatever is being raised to that exponent. All right? When we get into exponents that are negative or zero or even exponents that are fractional, and algebra two, that's when things can get a little more confusing to put it mildly. But for right now, positive integer exponents. Pause the video now if you need to because I'm going to scrub out some text. All right, here we go. Okay. Exercise two. Write out what each of the following terms involving exponents means as an extended product. Consider carefully your order of operations and remember that exponents come before multiplication. Now notice this didn't say it didn't say that we had to actually evaluate in any way. We just want to make sure we understand what this means. So for instance, X to the third, also known as X cubed, is simply the same as X times X times X so that's it. That's all I'm asking for. Likewise, X squared Y to the fourth really just means X times X times Y times Y times Y times Y, right? It's all it means. Now, this is where it might start to get a little confusing. Two X squared. You know, really all this means is that we've got this thing times this thing, which means we've got two X times two X that's it. Nothing more than that. Likewise, here, let's not be let's not be confused. Okay, the exponents come before the multiplication by four. So we don't get four X times four X times four X times four X we get four times X times X times X times X times Y times Y times Y but that four only shows up once. That should be contrasted with this problem. In this problem, what we really have is 9 X squared times 9 X squared times 9 X squared. Then if we really want to take this to the extreme, that's 9 times X times X times 9 times X times X times 9 times X times X okay, no simplified here. No simplifying. And the same song second verse here, right? Here we've got negative four X third times negative four X to the third because we're squaring and that just says, take the stuff inside and multiply it by itself once. Likewise, we can then multiply extend this out, negative four times X times X times X times negative four times X times X times X just so that we really see what's going on here with the repeated multiplication. So again, keep in mind your order of operations exponents come before multiplication. All right, unless there's parentheses that kind of superseded. Okay? All right, pause the video now if you need to. All right, I'm going to scrub. Let's keep going. Okay, number three, write out each of the following products and then express them in the form X to the N all right. So what does that mean? Let's take a look, letter a X squared times X cubed. Well, X squared is X times X X cubed is X times X times X, but all of this is 5 X's multiplied together. So that's X to the 5th. That's what they mean by writing it out as X to the N why don't you go ahead and do letter B and letter C okay? All right. Well, it may be a little bit obnoxious because of all the X's, but here we go. X to the 5th. X times X times X times X times X that's this guy. Times X times X that this guy, but then we just have X to the 7th. One, two, three, four, 5, 6, 7. X to the 7th. Finally, X times X times X times X and that's X to the fourth, but then I have to multiply it by X to the fourth again. And that, of course, gives me one, two, three, four, 5, 6, 7, 8, X to the 8th. I sense a pattern. So there must be something going on here. I'd like to be able to do this but I don't have to write all the X's out, right? So what happens when we multiply two quantities that have the same base, okay? Just to make sure we got it. This is called the base, and it's the same, and then these two are the exponents. Know your terminology. The base is the quantity, whether it's a variable or constant, doesn't matter. The base is the quantity that's being raised to the power. So I'd like you to pause the video for a second. Look back at exercise three. And think about what you can write down at the end of that. X times a, or sorry, X to the a times X to the B what does that equal? All right, let's go through it. Well, going back to exercise three really quick notice. X to the second times X to the third gave me X to the 5th. How does the 5 relate to the two and the three? Well, two plus three is 5. Let's see if it still works. Here I've got X to the 5th times X to the second. 5 plus two is 7. Here it is. Right? Here I've got X to the fourth times X to the fourth. Four plus four is 8. And there it is. So it appears that when we multiply two things that have the same base, we add their exponents. X to the a times X to the B equals X to the a plus B good rule, very, very, very important exponent rule. All right. So pause the video now if you need to, yes, I'm going to scrub out the text. All right, here we go. Last track, my cursor there for a second. All right, I'll scrub down. So we should now be able to use that exponent rule to very, very quickly simplify products like these, right? So for instance, X to the fourth times X to the 9th will be X to the four plus 9, and that'll be X to the 13th. Why don't you go ahead and simplify B and C really quickly? All right, let's go through them. Simple enough. This will be X to the two plus three plus four, that will be X to the 9th. Here, we don't have X, we have Y, but that's okay. This will be wide at the two plus 6. So that'll be Y to the 8th. It's easy. When we multiply two things that have the same base, we add their exponents. It doesn't matter whether the base is a variable, like X or Y, or whether the base is a numerical one. For instance, if I had two to the 5th, and I was multiplying it by two to the 11th, that would be the same as two to the 16th. It really doesn't matter. If I had X to the Y times X to Y I have X to the Y plus Y or X to the two Y all right? We always add the exponents when we multiply two things with the same base. The key there is they've got to have the same base. If I have something like X to the second times Y to the 5th, I do not get X, Y to the 7th. That doesn't make any sense. All right. So I'm going to clear out the text. Pause the video if you need to. All right, here we go. Okay, so back to justify. I know how much students love justifying. But let's do it. I want to justify multiplying these two. Monomials together. This one with this one. Using properties and exponent loss. So here we go. Let's take a look. 5 X to the third times two X to the 7th, all of a sudden, I flip flop, the X to the third and the two. Why can I do that? What property or reason can I give for it? Think about it for a minute. All right. Well, we can do that because multiplication is commutative. That would actually be okay to write down. You could say because multiplication is commutative. I'm going to write down commutative property. Of multiplication. In other words, I can flip flop, the X times two with two times I'm sorry, X to the third times two with two times X to the third. Now I'm going to take this and I'm going to group the 5 and the two together and the X to the third X to the 7th together. What property is that? All right, hopefully. You said it was the associative property. And again, it's the assertion of property. Of multiplication. All right, finally, here we're just saying X to the third times X to the 7th is X to the tenth, we'll call that. Exponent rule. Number one, exponent rule number one. There we go. Now, obviously we're going to be doing this a lot quicker and we'll get some practice with it in the next exercise. But I always want you to be thinking about this and I'm going to walk you through a lot of it when I explain it. Like all these properties that were really using. So pause the video now if you need to. All right, here we go. Okay, rewrite each of the following is equivalent expressions in simplest exponential form. So again, let me just walk you through this. When I rewrite this, what I'm going to do is I'm going to use these associative and commutative properties of multiplication to really think of this as two times 8 and X to the 7th times X to the 5th. Now many of you won't write out the step two times 8, then we'll be 16, X to the 7th times X to the 5th. Add those exponents and we get X to the 12th. Likewise, in letter B, I can do negative four times positive two. Sorry. Uh oh. I just got out of my presentation. I don't know how that happened. Let's go right back into it. Yay. You just got to see the inner workings of my Macintosh. I know, it's a little bit scary. We'll live with it. So turn to do exactly what we did in the last problem. We have negative four times two, group those together. And group the X cubed times X squared together. Negative times positive gives me a negative. Don't forget, negative 8, and then X cubed times X to the second. Will be X to the 5th. Finally, letter C is something that often when it pops up, kids can get confused by. Now, what do I do? What do I square? What do I do with that? Do I do the cubing first? Swearing first, maybe I add the three and the two and get 5. No. Remember, take it at its base level. What this really means is that I have this, right? With negative 6 X to the third inside of one of them and negative 6 X to the third inside of the other. This means I can now group the negative 6 times negative 6 together and the X to the third times X to the third together. Look what I'm really doing is I'm effectively squaring both parts of this term. All right, negative 6 times negative 6 is a positive 36, X to the third times X to the third. Is X to the 6th. All right. And that's that. Okay, pause the video now if you need to. Okay, here we go. All clear. All right. So exponents is repeated multiplication. Exponents are very important. They are the way that we summarize repeatedly multiplying by the same number or the same variable again and again and again. All right? There are going to be instrumental in many of the different things that we work with this year. So make sure that you feel comfortable with them. All right. Let me thank you for joining me for another common core algebra one lesson by email instruction. My name is Kirk weiser. And until next time, keep thinking. And keep solving problems.