US3 EXPONENTS
This eighth grade mathematics lesson focuses on calculating with exponents. It is the first lesson in a sequence of 12 lessons on this topic. The lesson is 50 minutes in duration. There are 36 students enrolled in the class.
| Time | Caption |
|---|---|
| 00:00:07 | I sit right here. |
| 00:00:14 | Okay. You guys, sit in the same proximity where you were sitting before. The only thing that's changed is- |
| 00:00:20 | Brian. |
| 00:00:21 | Would be this section right here. Ian, grab the second seat right there. Okay. |
| 00:00:29 | That- yeah, that can work the same for you. |
| 00:00:31 | It's on. |
| 00:00:37 | You got no homework? |
| 00:00:55 | Okay. We're starting a new chapter today, so Chris is gonna pass out your chapter assignments. |
| 00:01:01 | Go ahead and put that away. |
| 00:01:09 | So we're missing- Britney's here. |
| 00:01:13 | Okay. Everyone's here. |
| 00:01:16 | Okay attendance is (inaudible). I guess I better put the lights back on. |
| 00:01:45 | Miss (inaudible), I have a (inaudible). |
| 00:01:46 | What? |
| 00:01:48 | My homework. |
| 00:01:49 | Okay. Okay, get your assignment and go ahead and put it away for now. |
| 00:01:57 | Oh my gosh (inaudible). The next assignment (inaudible). |
| 00:02:03 | I don't know if you want their assignment sheet, but this is what we're gonna work on in class for a little bit today. Okay. |
| 00:02:09 | Miss (inaudible), are we- Do we need paper? (inaudible) |
| 00:02:11 | Yes. Go ahead and keep it for later. |
| 00:02:15 | Yes? |
| 00:02:16 | Do you have some paper? |
| 00:02:18 | Well, we- we'll worry about that later. Okay. |
| 00:02:21 | Okay. |
| 00:02:22 | Yes, Austin. |
| 00:02:23 | Do you want us to put our folders up (inaudible) put it in? |
| 00:02:25 | I just want you to put your assignment in your folder and put your folder away. |
| 00:02:28 | I will give you a sheet that we're gonna work on today that you can take notes on. |
| 00:02:31 | Okay. |
| 00:02:32 | Okay. The only thing you need on your desk is this piece of paper I'm gonna give you and a pencil. |
| 00:02:38 | Do we need a calculator? |
| 00:02:39 | You don't need your calculator right now. No. |
| 00:02:42 | Want our book? |
| 00:02:44 | You don't need your book right now either. |
| 00:02:46 | Send that five to that other group back there. |
| 00:02:48 | I forgot my book at home. |
| 00:02:50 | I'm a bad girl. |
| 00:02:52 | Yes, Tim? |
| 00:02:54 | Please excuse, tardy. Okay. |
| 00:03:02 | Where do I sit? |
| 00:03:03 | Right there. |
| 00:03:04 | Right there. |
| 00:03:20 | Okay. I have you broken into groups today because I want you working- |
| 00:03:25 | We're gonna work on this individually and we're gonna work on this as a group. |
| 00:03:29 | I do not want you to begin anything on the assignment, so turn the paper over please. |
| 00:03:35 | You didn't get one? |
| 00:03:40 | Yeah. He has one. |
| 00:03:41 | He has one? |
| 00:03:42 | Yeah. |
| 00:03:43 | Okay. So turn it over, and most of chapter eight that we're gonna deal with, is dealing with exponents. |
| 00:03:50 | And for exponents we have to remember back from what I first introduced to you in fifth grade. |
| 00:03:56 | You learned about two squared and two cubed. Okay. |
| 00:04:00 | Remember on the- the- the composition of your exponent it always has a base number. |
| 00:04:05 | And then the power or the exponent is the number that it is risen to. Okay. |
| 00:04:09 | The exponent stands for the number of times that the base is going to be multiplied. |
| 00:04:16 | So if we have two cubed, this tells me that the base is gonna be multiplied three times. |
| 00:04:21 | Two, times two, times two. |
| 00:04:24 | And the thing we're gonna learn about in this unit is exponential growth. |
| 00:04:28 | If we look over here, we have two cubes. This would be like two to the first power. |
| 00:04:33 | So if we made it two squared, which would be two times two, we would see that it grows to two squared. That's two times two, right? |
| 00:04:43 | Two cubed is two, times two, times two. Two to the third power. You see it's starting to double in size. |
| 00:04:52 | Then if we go two to the fourth, you're looking at... |
| 00:04:58 | And you see how quickly it starts to grow. Okay. |
| 00:05:04 | Now two to the fourth is how much? |
| 00:05:07 | Sixteen. |
| 00:05:08 | Sixteen. |
| 00:05:10 | Yeah. That's (inaudible). |
| 00:05:11 | See if I can get this to stand here. I don't know if I will. |
| 00:05:13 | Okay. So two to the fifth would be how much? |
| 00:05:16 | Twenty-five. |
| 00:05:17 | Twenty-five? |
| 00:05:18 | No. |
| 00:05:19 | Twenty. |
| 00:05:20 | Thirty-two. |
| 00:05:21 | Well, we know that- |
| 00:05:22 | Thirty-two. |
| 00:05:23 | Two to the fourth is 16. |
| 00:05:24 | Thirty-two. |
| 00:05:25 | And we take that and multiply it by two and we get? |
| 00:05:27 | Thirty-two. |
| 00:05:28 | Thirty-two. Okay. |
| 00:05:30 | So you see it is continually doubling. Look at the growth. |
| 00:05:35 | It's very big. |
| 00:05:36 | It's very, very big. |
| 00:05:38 | Okay. So whenever we're dealing with powers... |
| 00:05:42 | (inaudible) |
| 00:05:43 | Kamikaze. Whenever we're dealing with powers, we're gonna see that the growth goes quickly. Okay. |
| 00:05:50 | So think about that in terms of a graph. We've looked at slope. Right? And we've always had a constant slope. |
| 00:05:57 | What do you think if we were gonna graph two to the X power? |
| 00:06:02 | Think about that in turns. We found that two to the one was two. |
| 00:06:06 | Two. |
| 00:06:07 | Two to the two was four. |
| 00:06:08 | Four. |
| 00:06:09 | Two to the three was? |
| 00:06:10 | Eight. |
| 00:06:11 | And two to the fourth was? |
| 00:06:13 | Sixteen. |
| 00:06:15 | What do you think that graph's gonna look like? |
| 00:06:18 | It- a curve. |
| 00:06:20 | Yeah. It's gonna be a curve. Think about that. |
| 00:06:25 | Two to the first power, we get just two. Right? |
| 00:06:29 | Then we go up to four. We go up to eight. We go up to sixteen. We get some significant growth. Okay. |
| 00:06:38 | Yes? |
| 00:06:39 | Become a parabola if you go to the negative side. |
| 00:06:42 | That's a good question. We're gonna explore that in this unit. Okay. |
| 00:06:46 | We're gonna start dealing with the negative exponents. |
| 00:06:49 | But first we're just gonna play around and there's certain rules that you have to learn when you deal with exponents. Okay. |
| 00:06:54 | And that's what you're gonna work on as a group and individually. |
| 00:06:58 | First of all, there's gonna be three different rules that we're gonna find out about when we multiply exponents. |
| 00:07:04 | And what I want you to do on your worksheet in front of you, we're gonna work on just section one right now. |
| 00:07:09 | But before we get there, I'm gonna go through the three different rules of multiplying exponents. Okay. |
| 00:07:15 | If we think about this, what does two squared mean? |
| 00:07:19 | Two times two. |
| 00:07:20 | That's two times two. Right? And what does two cubed mean? |
| 00:07:24 | Two times two times two. |
| 00:07:25 | Two times two times two. |
| 00:07:28 | Okay. Now if I want you to write this out as a solution, we could go two times two is four times two. |
| 00:07:34 | But I want you to write it out as a solution of a power. What would this equal? |
| 00:07:41 | Two to the fifth power. |
| 00:07:42 | Two to the fifth power. Okay. |
| 00:07:45 | Then let's look at this. What is this telling me now that I have parentheses? |
| 00:07:49 | (inaudible) |
| 00:07:52 | Well- |
| 00:07:53 | You have to multiply it by- |
| 00:07:54 | You have to do (inaudible). |
| 00:07:55 | Two to the third power times two to the- |
| 00:07:59 | Third power. |
| 00:08:00 | Third power. Right? |
| 00:08:02 | We have because it's squared- |
| 00:08:03 | What? |
| 00:08:04 | Oh, squared. |
| 00:08:05 | We're taking the inside and we're multiplying it twice. Right? |
| 00:08:08 | Yes, Lucia? |
| 00:08:09 | Is- is that the same thing as doing two to the third power and then just, like, getting the answer- |
| 00:08:18 | Right now I'm not- in terms of finding out what the actual value is. I'm trying to find out what the power would be. Okay. |
| 00:08:24 | So I'm saying here I have two to the third power and it's squared. |
| 00:08:29 | So I have to take two to the third power and multiply it by itself twice. Right? |
| 00:08:33 | Mm-hm. |
| 00:08:34 | Well, we can even break this down further and what is this? Two- |
| 00:08:38 | Times two. |
| 00:08:39 | Two times two. And this is? |
| 00:08:41 | Two times two times two. |
| 00:08:42 | Two times two times two. |
| 00:08:43 | And what do we get here if I write it as a final power? |
| 00:08:46 | E:00] |
| 00:08:48 | Two to the sixth power. Okay. |
| 00:08:52 | Now for the last example, what's being squared here? |
| 00:08:56 | The X. |
| 00:08:57 | One X. |
| 00:08:58 | The two. |
| 00:08:59 | Two X. |
| 00:09:00 | Two and X or just- |
| 00:09:01 | Two. |
| 00:09:02 | Two. |
| 00:09:03 | Two. |
| 00:09:04 | Both of them. This is- |
| 00:09:05 | Two X. |
| 00:09:06 | Two X. |
| 00:09:07 | Two times X times two times- |
| 00:09:08 | X. |
| 00:09:09 | X. Or because we have the associative property in math, we go two times two, and X times X. Right? |
| 00:09:16 | Oh. |
| 00:09:17 | Is that any different? |
| 00:09:18 | Yeah. |
| 00:09:19 | No. |
| 00:09:20 | No. |
| 00:09:21 | No. |
| 00:09:22 | Okay. And this is two squared times- |
| 00:09:23 | X squared. |
| 00:09:24 | X squared |
| 00:09:25 | X squared. Right? |
| 00:09:27 | Okay. Turn your paper over. |
| 00:09:29 | Look at section one. For the next minute I want you to do the first three problems. |
| 00:09:35 | And in your head think if you can come up with the rule, after you've done these problems, for multiplying these exponents. |
| 00:09:44 | See if you see a pattern developing on your own. |
| 00:09:47 | Don't talk to your neighbor. Just do those three problems. |
| 00:09:56 | And if you need to expand it out to get the answer like I did, that's what I'd like you to do. |
| 00:10:01 | Expand the problem out, the multiplication out, and then combine it with an answer to a power. Okay. |
| 00:10:09 | Oh. |
| 00:10:26 | Okay. Expand that out. |
| 00:10:29 | Expand it out on your paper. |
| 00:10:58 | Okay. Expand that out. |
| 00:11:00 | Show me what that is. I want you to expand those out. All of those out before you come up with your answer. Okay. |
| 00:11:07 | Expand those out like I did on the board please. |
| 00:11:11 | Can I (inaudible) by expanding? |
| 00:11:12 | Right. |
| 00:11:13 | Yes. I want you expanding them out. Good. |
| 00:11:14 | Okay. |
| 00:11:15 | Mm-hm. And then give us the final answer. |
| 00:11:17 | So you want us to just tree, branch off, and just- |
| 00:11:19 | Yeah. |
| 00:11:20 | Okay. |
| 00:11:23 | Yes, Brett? |
| 00:11:26 | Just asking am I doing it right? Is that right? |
| 00:11:29 | Yes. |
| 00:11:30 | Yes? |
| 00:11:31 | Okay. Now that most of you have the problems worked out, I want you in your group to discuss if you see something- |
| 00:11:40 | A pattern developing when you're multiplying the exponents. |
| 00:11:43 | If you see something that you can do with the exponents. |
| 00:11:47 | Discuss it with your group. |
| 00:11:50 | Do we- do you want us to do our rule thing too? |
| 00:11:51 | Yes. |
| 00:11:53 | Okay. |
| 00:11:55 | I already know- what? |
| 00:11:57 | Yeah. Don't you, like, add the little exponents together to get (inaudible). |
| 00:12:01 | No (inaudible). |
| 00:12:02 | Otherwise, you just- |
| 00:12:04 | No. You're not (inaudible). |
| 00:12:05 | Okay. |
| 00:12:06 | Well, yeah that too, but what did you guys get for the rule? |
| 00:12:07 | Did you come up with a rule? |
| 00:12:08 | You just add all the exponents, and if there's a letter- |
| 00:12:09 | I- I understand. |
| 00:12:11 | Okay. So then what- what- that doesn't show me a rule, though. |
| 00:12:15 | What does A to the M, times A to the N. |
| 00:12:18 | If you say you're gonna add exponents, how would you write that? |
| 00:12:21 | A plus N equals... equals... |
| 00:12:28 | Okay. What's your base? Remember your exponent has to be written to a base. What's your base in this problem? |
| 00:12:34 | A. |
| 00:12:35 | Okay. |
| 00:12:36 | Okay. Did I do this right? |
| 00:12:38 | And so you're saying then that you just add the exponents. Okay. Very good. |
| 00:12:48 | You would say, add the exponents- |
| 00:12:49 | The exponents to- |
| 00:12:50 | Add the exponents to find your base... |
| 00:12:54 | Okay. So you're just saying that you're gonna- yeah. |
| 00:12:55 | To find your base exponent? Or how would you say, add the- add the exponents (inaudible). |
| 00:13:00 | How would you- what would you call it? |
| 00:13:01 | Add the exponents is sufficient. |
| 00:13:02 | Add the exponents. |
| 00:13:03 | Mm-hm. Mm-hm. |
| 00:13:04 | So you have to add the exponents? |
| 00:13:05 | So then show me that. |
| 00:13:06 | (inaudible), you know how to spell it, right? |
| 00:13:07 | Okay. Robert, what did you find out? |
| 00:13:10 | What- oh, I was gonna ask you a question. |
| 00:13:12 | Okay. Ask the question. |
| 00:13:14 | Do we have to write it in words? |
| 00:13:16 | No. You can- you can do it in symbols, Robert. |
| 00:13:18 | Okay. In that case, we are- |
| 00:13:20 | If I A to the M times A to the N, what did we find out when we multiplied? |
| 00:13:26 | A to the M plus N. |
| 00:13:29 | Okay. |
| 00:13:31 | Did anybody get something different? |
| 00:13:33 | (inaudible) |
| 00:13:34 | Okay. What did you get, Chad? |
| 00:13:35 | A to the N in the parentheses and then N on the outside of the parentheses. |
| 00:13:39 | No that's not right. |
| 00:13:40 | No. |
| 00:13:41 | A to the M in parentheses and to the outside N. Okay. |
| 00:13:46 | So then you found- so how would I do it if I had A to the two to the four? |
| 00:13:54 | You would A to the sixth? So you just add it? |
| 00:13:57 | Okay. So you're saying that if you have it this way, it's being added. Okay. |
| 00:14:01 | What does parentheses stand for in math? |
| 00:14:04 | Multiplication. |
| 00:14:06 | Okay. So would someone seeing this say, oh, multiplication or addition, do you think? |
| 00:14:10 | Multiplication. |
| 00:14:11 | Multiplication. |
| 00:14:12 | Multiplication. Okay. So then how are you gonna show me since you tell me that if I add these two that's gonna be the answer. |
| 00:14:18 | How can I show it such that it's being added? |
| 00:14:22 | What Robert says. |
| 00:14:23 | Okay. Okay. |
| 00:14:26 | So are- yes, Austin? |
| 00:14:27 | I got A and then 27. |
| 00:14:31 | On which problem? |
| 00:14:32 | On the rule. |
| 00:14:33 | For the rule. |
| 00:14:34 | Okay. And what did you plug in for M and N, then? |
| 00:14:38 | I said N is A to N to the A B C's is 13. |
| 00:14:43 | Okay. So you took all- you took all the- you took all the A's out of the problem. Right? |
| 00:14:47 | Yeah. |
| 00:14:48 | Okay. Those were just examples to look at to find a pattern. |
| 00:14:51 | 'Cause if you look at the first problem, we had A square times A to the fourth and we got A to the- |
| 00:14:57 | Sixth. |
| 00:14:58 | Sixth. And two plus four is- |
| 00:15:00 | Six. |
| 00:15:01 | Six. |
| 00:15:02 | Okay. Then we had A squared times A. And what's the exponent on A? |
| 00:15:05 | One. |
| 00:15:06 | One. |
| 00:15:07 | So two plus one is- |
| 00:15:08 | Three. |
| 00:15:09 | And we got A cubed. Right? For number two. |
| 00:15:10 | Yeah. |
| 00:15:11 | And then on number three we had A cubed times A times A to the fourth. So three plus one plus four is- |
| 00:15:18 | Eight. |
| 00:15:19 | A to the eighth. |
| 00:15:20 | Okay. So you found that when we are multiplying the same base, did you notice that the base had to be the same? |
| 00:15:29 | Okay. When you multiply the same base with exponents we just add the exponents to get the new answer. Right? |
| 00:15:40 | Okay. Let's look at section two. Do problems four through six. Multiply that out and see if you can find the rule for that. |
| 00:15:48 | And I want you to expand it. It's very important that you expand it. |
| 00:15:51 | 'Cause when you expand it, you'll be able to see the pattern much more quickly. |
| 00:15:56 | Okay. |
| 00:15:59 | Three A, two, two. |
| 00:16:00 | Miss (inaudible)? |
| 00:16:01 | Yes. |
| 00:16:02 | When you do this three, you break that? |
| 00:16:04 | Okay. That's cubed. Right? |
| 00:16:06 | So how many times would you multiply the inside part? What's in the inside part? |
| 00:16:09 | The two A squared. |
| 00:16:11 | Okay. So write A squared. |
| 00:16:13 | And how many times would you have A squared? |
| 00:16:15 | Three- oh, three times. |
| 00:16:16 | Oh, three times. |
| 00:16:17 | Okay. So write that out. |
| 00:16:18 | (inaudible), you would break it down, and- right. |
| 00:16:19 | Okay. So now you have that. Can you break that down even further now? |
| 00:16:23 | I did. |
| 00:16:24 | What's A squared look like? |
| 00:16:25 | A times A. |
| 00:16:27 | Okay. And then you have to have how many sets of those? |
| 00:16:28 | A multiplied by A. |
| 00:16:29 | Three. |
| 00:16:30 | Three. |
| 00:16:31 | Three. |
| 00:16:32 | Three. So then you write that out and you've expanded it. |
| 00:16:34 | It'd be A, by A, by A, by A, by A, by A. |
| 00:16:35 | So it would be A to the sixth. |
| 00:16:37 | So how many A's would you have? |
| 00:16:39 | Six. |
| 00:16:40 | A to the- |
| 00:16:41 | So it would be A to the- |
| 00:16:42 | Sixth power. Good. Okay. So make sure you expand it out. |
| 00:16:43 | That's- |
| 00:16:44 | That's what I got. |
| 00:16:45 | Yes. Lucia, did you have a question? |
| 00:16:47 | Oh, I get it. I get it. |
| 00:16:48 | No, I was asking the same thing as she did, but I heard you explain it. |
| 00:16:50 | What's that? |
| 00:16:51 | How you- I'm gonna just write it three times, write this three times? |
| 00:16:56 | What's in the parentheses is being cubed. So it's three times. Right? |
| 00:16:59 | Yeah, so- |
| 00:17:00 | Okay. Then how do you break out for your A squared? |
| 00:17:03 | Two A's? |
| 00:17:04 | You go two A's. Right? |
| 00:17:05 | Yeah. |
| 00:17:06 | A times A, and you do that three times. Right? So how many A's do you end up with? |
| 00:17:09 | A to the sixth. |
| 00:17:10 | Six. |
| 00:17:11 | A to the sixth. |
| 00:17:12 | So then it'll be A to the sixth. He got A to the fifth. |
| 00:17:14 | How did you all get six A's? I got five A's. |
| 00:17:15 | Because two times three is six. |
| 00:17:16 | Did you do- okay. |
| 00:17:17 | And that's (inaudible) six. |
| 00:17:18 | Okay. Show him. |
| 00:17:19 | So see, you do. |
| 00:17:20 | Well now show him, Lucia. Show him how yours is broken out. Explain to Chad how that works. |
| 00:17:24 | Here. If there's three, it's A squared times three, so you have to do three A squared, then you have to break it down to A times A, |
| 00:17:33 | 'cause that's what A squared is, and then you do that for each of them and that's three A squared with (inaudible). |
| 00:17:34 | So basically I- isn't the rule, like, you just times whatever's in here? The number by that. |
| 00:17:38 | So you're gonna multiply your exponents, right? |
| 00:17:40 | Yeah. |
| 00:17:41 | Okay. Did you guys have it? |
| 00:17:42 | So you multiplied the two and three and got six? |
| 00:17:43 | Yep. |
| 00:17:44 | Yeah. |
| 00:17:45 | You have your rule? |
| 00:17:46 | (inaudible) Oh. |
| 00:17:47 | What's your rule? Don't forget your rule. |
| 00:17:48 | Don't forget to do your rule if you have determined it with your group. |
| 00:17:51 | All right. |
| 00:17:52 | Yes. |
| 00:17:54 | Can we sh- we (inaudible) that it was multiplying. Can we show it anyway we want? |
| 00:17:56 | Okay. That's- that's exactly- yes. |
| 00:18:00 | Hi. I'm on TV. |
| 00:18:06 | Yes. (inaudible) |
| 00:18:07 | How do you do this one? It said like- |
| 00:18:10 | So on number five we (inaudible)? |
| 00:18:12 | Okay. So you have A squared. Right? |
| 00:18:14 | Yeah. |
| 00:18:15 | And how many times do you need A squared? |
| 00:18:17 | Three. |
| 00:18:18 | Three times. So you have it once here. Where's the other two times? |
| 00:18:22 | Don't you need three A squares? |
| 00:18:23 | This one. |
| 00:18:24 | No. No. No. Don't look at that bottom part. |
| 00:18:26 | Oh. |
| 00:18:27 | I want you to write that out in expanded form. |
| 00:18:29 | What's A squared cubed? What does that look like? |
| 00:18:33 | Isn't it- She says that, like- Didn't you do it this way? |
| 00:18:38 | No. No. No. |
| 00:18:39 | That's what... |
| 00:18:40 | What- what's being cubed here? |
| 00:18:41 | Three. |
| 00:18:42 | No. What's being cubed here? |
| 00:18:43 | Oh. |
| 00:18:45 | What's in the parentheses? |
| 00:18:46 | Three twos. |
| 00:18:47 | Okay. So write three A twos for me. |
| 00:18:50 | Three A twos. |
| 00:18:51 | Oh. |
| 00:18:52 | 'Cause isn't that- don't you have A squared three times? |
| 00:18:53 | Oh, I get it. |
| 00:18:54 | No. Write three A twos out. |
| 00:18:57 | Three A twos. |
| 00:19:00 | Isn't this A two A squared times A squared times A squared? |
| 00:19:01 | Oh. Oh. |
| 00:19:04 | So this is wrong? You have A squared, A squared, like that? |
| 00:19:09 | Okay. Now can you break those down further? |
| 00:19:17 | So it's A- six |
| 00:19:19 | Okay. And redo these. You guys, okay- with the same logic and see if you see a pattern. |
| 00:19:24 | Okay, Mrs. Scott? |
| 00:19:25 | Yes. |
| 00:19:26 | It's A to the sixth power. |
| 00:19:27 | I don't understand that. I don't- |
| 00:19:29 | Okay. So break those down. What's A cubed look like? |
| 00:19:33 | That, I don't know. |
| 00:19:35 | What's A cubed look like? How do you write A cubed in expanded form? How many A's is it? |
| 00:19:39 | Three. |
| 00:19:41 | So write A times A times A. |
| 00:19:45 | Okay. |
| 00:19:46 | And then what do you have here? Another A cubed. So what do you write? |
| 00:19:49 | Another? Where did you get another? |
| 00:19:50 | There's A cubed times A cubed. I was just reading your paper. |
| 00:19:53 | Oh, well- |
| 00:19:54 | That's right. |
| 00:19:55 | Oh. |
| 00:19:56 | So write out another A cubed. Another A cubed. Break it out. |
| 00:20:00 | Oh, we have to keep going? So we can't just stop right there. |
| 00:20:03 | So how many A's do you have? |
| 00:20:04 | Well, how- what about the two? Where did we get the two from? |
| 00:20:07 | Look it. What's being squared? A to the three. |
| 00:20:10 | Yeah. |
| 00:20:11 | So A three times A three, is that- |
| 00:20:12 | Oh, twice. Huh? |
| 00:20:13 | Yeah. Now you're breaking these down, A times A times A. So how many A's do you have? |
| 00:20:18 | Six. |
| 00:20:19 | There you go. Okay. |
| 00:20:20 | Do we really have to break them down the whole way? |
| 00:20:23 | Mm-hm. 'Cause if you can pick it up, yes. |
| 00:20:25 | Okay. Let's look at number four. |
| 00:20:35 | Okay. Are we all focused? Let's go. |
| 00:20:36 | Yeah. |
| 00:20:37 | Number four. What did you get, Lucia, in your group? |
| 00:20:41 | (inaudible) |
| 00:20:42 | No. I just want to know what the answer to number four is. |
| 00:20:45 | A to the sixth. |
| 00:20:46 | A to the sixth. Okay. Now what did you get over here, Phoebe, for number five? |
| 00:20:49 | A to the sixth. |
| 00:20:50 | And Robert's group, what did you get for number six? |
| 00:20:53 | A to the eighth. |
| 00:20:54 | A to the eighth. Okay. Did someone find a rule? |
| 00:20:57 | I know. I know. I know. |
| 00:20:58 | Chris, what did you find out? |
| 00:21:00 | A to the M times N, or A to the MN. |
| 00:21:03 | Okay. When we have an exponent within parentheses raised to an exponent, we get A to the what? |
| 00:21:12 | A to the M times N, or A to the MN. |
| 00:21:15 | A to the M times N. |
| 00:21:17 | So he's saying, his rule is that you're going to multiply the exponents. Does that work? |
| 00:21:24 | A squared cubed, two times three is six. Right? |
| 00:21:27 | Mm-hm. |
| 00:21:28 | And if we look at number six, two times four is eight. Okay. |
| 00:21:31 | So now you've got two rules already down. You've discovered them all by yourself. I didn't have to tell you them. Right? |
| 00:21:37 | You found that when you multiply exponents that have the same base, we're gonna add them. |
| 00:21:43 | Then when we take an exponent and raise it to a power, we're gonna multiply those exponents. Okay. |
| 00:21:51 | Let's look at the third section. Okay. |
| 00:21:54 | Now we have two terms within the parentheses being raised to a power. Okay. Expand that out just like I did down here. |
| 00:22:05 | Two X, we went two X times two X, and then we grouped our like terms, two times two, we got two squared times X squared. Okay. |
| 00:22:13 | Do that with seven, eight, and nine and see if you can come up with the rule. |
| 00:22:28 | We- we're having trouble with the update for R T A (inaudible). |
| 00:22:40 | I'm trying to work something (inaudible). |
| 00:22:41 | Then we'll let them know 'cause I've- I already got- |
| 00:22:48 | Miss? |
| 00:22:50 | And then A five times A five. And then A four times A four. |
| 00:22:55 | Okay. |
| 00:22:56 | You just put the exponent after the number. |
| 00:22:57 | You don't- you don't need to get in- get in on it. Just let it- let them go. |
| 00:23:00 | Okay. |
| 00:23:01 | So if we did this, Could- A three times A three will be the same as A B three. Right? |
| 00:23:07 | I don't know. Write that down, what you said you think it is. |
| 00:23:11 | What? |
| 00:23:12 | Okay. That's A cubed times B cubed. Right? |
| 00:23:15 | A, A, A, times B, B, B. |
| 00:23:18 | Okay. So you asked me is it the same as what? |
| 00:23:22 | A B. Okay. So, it would be A B times A B times A B (inaudible). |
| 00:23:28 | Wanna ask her like how do you combine it. How do you combine it? |
| 00:23:29 | Okay. |
| 00:23:30 | Those aren't the same. |
| 00:23:31 | Yeah they're the same. How many A's do I have? |
| 00:23:35 | Three and three. |
| 00:23:36 | Okay. So I'm okay. |
| 00:23:37 | But you- I though you asked me is it the same as this. A B cubed. |
| 00:23:42 | Is it? It is? |
| 00:23:44 | No. |
| 00:23:45 | That says to me A times B times B times B. |
| 00:23:51 | A three, B three. |
| 00:23:53 | That's- that's what you have to make sure. Okay. |
| 00:23:56 | 'Cause this is when you asked me- |
| 00:23:57 | Right. I told you. |
| 00:23:59 | That's what it sounded like you had asked. And that's not the same as that. And you understand why. Right? |
| 00:24:02 | Okay. Yes? |
| 00:24:03 | So this would be right, and (inaudible). |
| 00:24:06 | Uh-huh. Okay. Try to develop your rule if you can, too, with your group? |
| 00:24:12 | How do you keep- You just combine them. Right? |
| 00:24:15 | Okay. Well, think about this. This is- you have A B cubed. |
| 00:24:19 | If I were to expand this- okay, if I was going to expand that, that would be A times B times B times B. Is that the problem I gave you? |
| 00:24:29 | No. |
| 00:24:30 | Okay. So find out where you went wrong with this logic. I don't see you writing this out in expanded form. |
| 00:24:36 | All right. |
| 00:24:38 | Okay? |
| 00:24:40 | Yes? |
| 00:24:41 | Am I doing this right? |
| 00:24:42 | Yes. |
| 00:24:43 | Oh. |
| 00:24:45 | Miss (inaudible)? |
| 00:24:46 | Yes. |
| 00:24:47 | I told you, brat. |
| 00:24:48 | I don't understand the next step. Okay. So far I (inaudible) A times B so that'd be A B. |
| 00:24:50 | Okay. |
| 00:24:52 | And then it's times three. |
| 00:24:53 | Three times. Yeah. |
| 00:24:54 | So A times- A B, A B, A B. What do I do next? |
| 00:24:56 | Okay. So how many A's do you have? |
| 00:24:58 | Three A's and three B's. |
| 00:24:59 | Three As. |
| 00:25:00 | And three Bs. |
| 00:25:01 | Okay. So what does this say? That says one A and three B's. |
| 00:25:03 | Is this right? |
| 00:25:04 | Oh. |
| 00:25:05 | One A and three Bs? |
| 00:25:06 | Doesn't that what that says? |
| 00:25:07 | Where? |
| 00:25:08 | Right here. Doesn't that say one A? A to the one times B to the three? |
| 00:25:13 | That's not what I have. I have this. |
| 00:25:14 | Oh, what? |
| 00:25:15 | Is it A cubed? |
| 00:25:16 | Well, would it be like- |
| 00:25:17 | So I don't know. How many A's do you have? |
| 00:25:18 | Three. |
| 00:25:19 | Three. |
| 00:25:20 | So what's- what's- |
| 00:25:21 | A. |
| 00:25:22 | Cubed. A cubed times (inaudible)? |
| 00:25:24 | A cubed times A cubed? |
| 00:25:26 | And that's- that's- |
| 00:25:27 | So then I was right? |
| 00:25:28 | Yeah. |
| 00:25:29 | Oh. Cool. |
| 00:25:30 | So A B times- to the third isn't right? |
| 00:25:31 | Would it be (inaudible) or would it be times that? |
| 00:25:34 | Is that right, Ms. Scott? |
| 00:25:35 | That's- that's fine. |
| 00:25:36 | A cubed times B cubed. |
| 00:25:37 | Uh-huh. |
| 00:25:38 | So A cubed times- |
| 00:25:39 | And- and then if you can get rid of the multiplication it can just be A cubed, B cubed. Can't it? |
| 00:25:46 | Sure. |
| 00:25:48 | Okay. |
| 00:25:55 | Yes, Phoebe? |
| 00:25:56 | Okay. I don't- okay, for number seven, wouldn't it equal, like- wouldn't you put in for instance A times A times A, |
| 00:26:03 | and then other times you'd put it B times B times B? |
| 00:26:05 | Okay. And what- that would equal what? Write it as powers. How many A's do you have? |
| 00:26:10 | A- or A cubed. |
| 00:26:12 | Okay. Times what? |
| 00:26:14 | Times B cubed. So would that be the answer or would I put A B cubed? |
| 00:26:19 | Okay. Write A B cubed for me. |
| 00:26:22 | Five times and then times B, times B, times B. |
| 00:26:24 | Okay. How many A's is that? |
| 00:26:26 | Three. |
| 00:26:27 | Is it? Where's the three on the A? |
| 00:26:28 | Oh. |
| 00:26:29 | Oh. So you'd have to put A cubed plus B cubed? |
| 00:26:33 | Are we- did I tell you to plus? |
| 00:26:35 | No. |
| 00:26:36 | What are we doing? |
| 00:26:37 | Times. |
| 00:26:38 | Multiplying. |
| 00:26:39 | Okay. And how do we designate multiplying in algebra? |
| 00:26:42 | Parentheses. |
| 00:26:43 | And what other way? |
| 00:26:44 | And the little dot thing. |
| 00:26:45 | And what other way? |
| 00:26:47 | Times, the X. |
| 00:26:48 | The X. |
| 00:26:49 | And what other way? |
| 00:26:50 | I don't know. |
| 00:26:52 | Well, if I write A B, what does that mean? |
| 00:26:55 | A times B. |
| 00:26:56 | So you can just have them next to each other, right? |
| 00:26:59 | Yeah. |
| 00:27:00 | And that still means multiplication. |
| 00:27:01 | And then would I put like A B cubed, or A cubed and B cubed. |
| 00:27:05 | Yep. |
| 00:27:06 | What? A cubed and B cubed. Is that what I would do? |
| 00:27:09 | Uh-huh. |
| 00:27:11 | Like- |
| 00:27:15 | I mean B cubed. Is that what I would put? |
| 00:27:17 | Uh-huh. |
| 00:27:18 | That's the answer. That's how I would get it? |
| 00:27:20 | I hope so. Keep working. See the other ones. |
| 00:27:22 | So we put it in parentheses. |
| 00:27:24 | Oh, that doesn't help. |
| 00:27:25 | Yes, you could also do it in parentheses. But we want you to get it expanded so that the powers are on each one of the variables. |
| 00:27:32 | Okay. Let's look at number seven. The quantity A B cubed. |
| 00:27:40 | What did we find out when we had the quantity of A B cubed? Robert, how would we rewrite that? |
| 00:27:46 | A B Cubed. |
| 00:27:47 | Okay. A cubed times B cubed. Right? |
| 00:27:49 | Mm-hm. |
| 00:27:51 | And on eight, Courtney, the quantity A B to the fifth power? |
| 00:27:57 | A to the fifth times B to the fifth. |
| 00:27:59 | Okay. And what about number nine? Ryan, you have that? A B- |
| 00:28:05 | It's A to the fourth and B to the fourth. |
| 00:28:07 | Okay. Did someone find the rule? |
| 00:28:09 | We did. |
| 00:28:10 | Blake? |
| 00:28:12 | It's A to M times B to M. |
| 00:28:15 | Okay. So when we have our base in here being raised to a power, each individual term is raised to that power. So we get- |
| 00:28:33 | A to the M, B to the M. Okay. |
| 00:28:39 | And remember when we have two variables next to each other, they're being multiplied. |
| 00:28:45 | Okay. So there's the third rule of exponents for multiplication that you need to know. Okay. |
| 00:28:52 | When we have the same base being raised to a power, we just add the powers. |
| 00:28:58 | When we have a base raised to a power, raised to a power, we multiply the exponents. |
| 00:29:04 | When we have two terms raised to a power within parentheses, we raise each term to that power. Okay. |
| 00:29:13 | So now, let's work on some division. Let's think about this. |
| 00:29:19 | We're going to expand this, so this is two, times two, times two, times two. And two squared is two times two. Okay. |
| 00:29:28 | What is two over two equal to? |
| 00:29:30 | One. |
| 00:29:31 | One. |
| 00:29:32 | One. And what is two over two equal to? |
| 00:29:35 | One. |
| 00:29:36 | One. So what are we left with? |
| 00:29:37 | Two. |
| 00:29:38 | We're left with one square, which is what? |
| 00:29:41 | Four. |
| 00:29:42 | One squared. |
| 00:29:43 | One, right? |
| 00:29:44 | One, yeah. |
| 00:29:45 | Okay. And two squared. Right? |
| 00:29:48 | So... |
| 00:29:49 | So... |
| 00:29:50 | What do we do with the one? |
| 00:29:51 | Okay. Shh. |
| 00:29:52 | Multiply it. |
| 00:29:53 | Okay. We take our two to the fourth power, we expand it out two times two, times two, times two. |
| 00:30:01 | Our denominator is two squared, so it's two times two. And we know that a number divided by itself is always one. |
| 00:30:10 | So this becomes one and this becomes one, and one times one is one. Right? And one times two is two. |
| 00:30:16 | So the ones there, we just don't have to write it. |
| 00:30:20 | So what are we left with? Two times two, or two squared. Right? |
| 00:30:24 | That's all you write? |
| 00:30:25 | Okay. |
| 00:30:27 | Now, what is this saying? This is saying I have four over two how many times? |
| 00:30:33 | Three. |
| 00:30:34 | Three times. |
| 00:30:39 | So how many fours do I have? |
| 00:30:41 | Three. |
| 00:30:42 | And how many twos do I have? |
| 00:30:43 | Three. |
| 00:30:45 | Oh, that's pretty easy. |
| 00:30:46 | Okay. |
| 00:30:47 | That's all we have to do? |
| 00:30:48 | So do section four and come up with a rule for me. Section four, problems 10 through 12. |
| 00:30:53 | Oh my gosh. I feel so stupid (inaudible). |
| 00:30:55 | How does this one look? For section (inaudible)? |
| 00:30:59 | A to the sixth for A squared. That's wrong. |
| 00:31:04 | Isn't it A- |
| 00:31:05 | How many A's. Look it. If these are ones. Right? |
| 00:31:07 | Mm-hm. |
| 00:31:08 | How many A's do you have left? |
| 00:31:09 | Four. |
| 00:31:10 | And he wrote three. |
| 00:31:11 | Oh. |
| 00:31:12 | I wrote three, too. |
| 00:31:13 | Okay. And look at these. A to the fourth over A to the one. A over A is one. Right? So how many A's would be left? |
| 00:31:21 | Two. |
| 00:31:22 | Three. |
| 00:31:23 | Two. |
| 00:31:24 | Three, cause right there. |
| 00:31:25 | Two. |
| 00:31:26 | One, two, three. |
| 00:31:27 | Yeah. |
| 00:31:28 | Three? |
| 00:31:29 | Yeah, cause there's three. |
| 00:31:30 | Help each other out. See if you come up with a rule on that one. |
| 00:31:33 | A to the fourth power. |
| 00:31:40 | How do we do (inaudible), like that? |
| 00:31:41 | Yes, Tiffany? |
| 00:31:43 | I don't understand how to do that one. |
| 00:31:45 | Okay. So how many A's do I need to write out here? |
| 00:31:47 | Four- six. |
| 00:31:48 | Six. |
| 00:31:49 | So write out A- six A's. |
| 00:31:51 | Okay. |
| 00:31:52 | This is... Cynthia? |
| 00:31:54 | Uh-huh? |
| 00:31:55 | Did you get A to third? |
| 00:31:57 | A to the fourth. Right? |
| 00:31:59 | Okay. And what's it being divided by? |
| 00:32:02 | A to the second. |
| 00:32:03 | Okay. So put your division line. And then how many A's are you gonna write? |
| 00:32:09 | Two. |
| 00:32:10 | Two. |
| 00:32:11 | Okay. |
| 00:32:13 | Now A over A becomes what? |
| 00:32:15 | One. |
| 00:32:16 | One. |
| 00:32:17 | And A over A becomes what? |
| 00:32:18 | One. |
| 00:32:19 | So- |
| 00:32:23 | Okay. And how many A's do you have left? |
| 00:32:25 | Four. |
| 00:32:26 | Four. |
| 00:32:27 | Okay. |
| 00:32:28 | So wouldn't it be A to the fourth? |
| 00:32:29 | So it's two A squared? |
| 00:32:30 | Why is it two A? What's one times one? |
| 00:32:32 | One. |
| 00:32:33 | One. |
| 00:32:34 | And what's one times A to the fourth? |
| 00:32:36 | A to the fourth? |
| 00:32:37 | A fourth. A fourth. |
| 00:32:39 | So then it's just A to the fourth. |
| 00:32:41 | He was the smart one. |
| 00:32:42 | 'Cause isn't that your multiplication? One times one is one. One times A is A. And A times A is A squared. And A times- |
| 00:32:49 | Oh, okay. |
| 00:32:50 | Okay. |
| 00:32:52 | Okay, I got it. |
| 00:32:53 | Yes? |
| 00:32:54 | Would you add these together, the four and the four (inaudible)? |
| 00:32:58 | Okay. If you have A to the fourth up top, why do we have A to the fourth on the bottom? |
| 00:33:02 | What's the denominator? |
| 00:33:05 | Oh, (inaudible). |
| 00:33:06 | How many A's are supposed to be down there? |
| 00:33:07 | Okay. |
| 00:33:08 | Okay. |
| 00:33:09 | Is this right, Miss Scott? Did you do A to the sixth and A to the (inaudible)? |
| 00:33:11 | Okay. |
| 00:33:12 | Is this A to the fourth? |
| 00:33:13 | Uh-huh. What's left here? |
| 00:33:15 | I still don't get how it's down by four. |
| 00:33:17 | Okay. What's this? A over A? |
| 00:33:19 | One. |
| 00:33:20 | Okay. And what's this? |
| 00:33:21 | One. |
| 00:33:22 | Okay. And what's this? |
| 00:33:23 | Four. |
| 00:33:24 | Okay. A to the fourth. Right? What's one times one? |
| 00:33:27 | One. |
| 00:33:28 | And what's one times A to the fourth? |
| 00:33:30 | A to the fourth. |
| 00:33:33 | Oh, I got the rule now. Okay. |
| 00:33:35 | You get- Does that make sense to you? |
| 00:33:36 | Yeah. |
| 00:33:37 | Miss Scott, is this right? Did you get A to the sixth on top and then A- |
| 00:33:39 | Squared on the bottom. Yep. |
| 00:33:41 | And then- |
| 00:33:42 | Yep. |
| 00:33:43 | Does the answer equal A to the... |
| 00:33:44 | And this one's A. Right? 'Cause- |
| 00:33:46 | Okay. And see if you can find the rule now. See if you can find the rule. |
| 00:33:50 | Okay. Let's look at problem- |
| 00:33:52 | You just add them. You just add them. |
| 00:33:55 | Problem number 10. |
| 00:33:57 | Dominic, what did you get for the answer to problem number 10? |
| 00:34:00 | A to the fourth. |
| 00:34:02 | A to the fourth. Okay. Sam, on number 11, what answer did you get? |
| 00:34:06 | A to the third. |
| 00:34:07 | A cubed. Okay. And on number 12. Chelsea, what did you get? |
| 00:34:12 | A. |
| 00:34:13 | A. Okay. Do we have a rule? We have a rule? |
| 00:34:17 | Lucia? |
| 00:34:18 | A to the M minus N. |
| 00:34:21 | A to the M minus N. |
| 00:34:23 | So you're saying that when I have a base to an exponent and it's the same base, we take the exponents- |
| 00:34:33 | We take the numerator and we subtract the denominator from it. Okay? |
| 00:34:38 | Yeah. |
| 00:34:39 | Does that make sense? |
| 00:34:40 | Yeah. |
| 00:34:41 | Okay. Very good. |
| 00:34:43 | So now you've got the fourth rule. We've only got one more rule to learn. Okay. |
| 00:34:48 | Now on this one, on this rule we need to take these and expand them out this way so that you can see the rule develop. Okay. |
| 00:34:57 | So go ahead and do 13 through 15 quickly. |
| 00:35:00 | Oh. |
| 00:35:03 | Yes? |
| 00:35:04 | I need help. How do you get it? |
| 00:35:06 | Oh yes, I do. |
| 00:35:07 | Okay. |
| 00:35:09 | Okay. So first you expand it out, right? You get A times A, times A, over B times B, times B. |
| 00:35:16 | So wouldn't it equal A over B? I don't know. |
| 00:35:20 | Well, how many A's do you have? |
| 00:35:21 | Three. |
| 00:35:22 | So how do you write that? |
| 00:35:23 | Oh, so it wouldn't be A cubed over B cubed? |
| 00:35:27 | Okay. Looks good to me. Try the next one and see if you see a pattern develop. |
| 00:35:32 | A times (inaudible). |
| 00:35:34 | That was really, really (inaudible). |
| 00:35:36 | B times B. |
| 00:35:38 | So now, what would that equal? Zero, or would it just equal A? |
| 00:35:41 | Yeah. |
| 00:35:42 | Yes. |
| 00:35:43 | Miss Scott, am I doing this right? |
| 00:35:44 | Miss Scott, on number 13, would it be A over B or would it just be zero? 'Cause they all- |
| 00:35:47 | Okay. Well, what- what is- You have A divided by B, right? |
| 00:35:51 | And what power is it being raised to? Okay. |
| 00:35:53 | To three. |
| 00:35:54 | Three. |
| 00:35:55 | So it's A over- |
| 00:35:56 | Third. So, it'd be, right? |
| 00:36:00 | Okay. |
| 00:36:01 | But those both canc- the thirds cancel out so it'd just be A over B? |
| 00:36:05 | They do. |
| 00:36:06 | Couldn't it be one? |
| 00:36:07 | How do you know that? |
| 00:36:08 | I don't know because they're- they're both to the third power. |
| 00:36:10 | Because- |
| 00:36:11 | Okay. But what's to the third power? |
| 00:36:12 | But you don't know. |
| 00:36:13 | A? |
| 00:36:14 | A and B. So it's still- So they're different. So it's just A over B. |
| 00:36:17 | Okay. Are they like terms? |
| 00:36:18 | Nn-hnh. |
| 00:36:19 | So can we combine unlike terms? |
| 00:36:21 | Nn-hnh. |
| 00:36:22 | No. So then they're in simplest form, then. Right? |
| 00:36:24 | Okay. Yeah. |
| 00:36:25 | Okay. |
| 00:36:26 | So what was the last answer if they're in simplest form? |
| 00:36:30 | It's A to the third, B to the third. |
| 00:36:31 | Why- |
| 00:36:34 | But Miss Scott, did you say- |
| 00:36:35 | Couldn't that still be A over B. though? |
| 00:36:36 | Ms. Scott? |
| 00:36:37 | Yes, I'm listening, Lucia. |
| 00:36:38 | You said it was (inaudible) like, this is all one. Didn't you, because- |
| 00:36:43 | Okay. So- So how many A's do you have? |
| 00:36:46 | Three. |
| 00:36:47 | Oh. |
| 00:36:48 | A cubed and B cubed. Right. |
| 00:36:49 | Okay. Now, let's say I had A on the bottom- to go to your point. 'Cause you were saying the threes cancel out. |
| 00:36:57 | Okay. What if I had A cubed over A cubed? What would I have? |
| 00:37:01 | Then that would cancel out and it would be these here. Right? Or no. |
| 00:37:05 | 'Cause they're both- |
| 00:37:07 | What's A- What- What's- What's A over A? |
| 00:37:09 | Oh. Oh. It would still be- |
| 00:37:11 | A over- |
| 00:37:12 | What's A over A equal? |
| 00:37:13 | One. |
| 00:37:14 | If I had three of those, what would I have? |
| 00:37:15 | Three. |
| 00:37:16 | One. |
| 00:37:17 | Well what's one times one times one? |
| 00:37:18 | A to the third. |
| 00:37:19 | One. |
| 00:37:20 | Oh. One. |
| 00:37:21 | One. |
| 00:37:22 | One, yeah. |
| 00:37:23 | One. |
| 00:37:24 | Three. |
| 00:37:25 | Okay. So if- if the denominator was the same variable as the numerator, yes, we could simplify it. |
| 00:37:29 | Mh-hm. |
| 00:37:30 | But you can't simplify it because it's a different term. Right? |
| 00:37:32 | Mm-hm. |
| 00:37:33 | Okay. Quickly now, let's go. Number 13. |
| 00:37:40 | What is our answer, Robert? |
| 00:37:43 | A to the third over B to the third. |
| 00:37:45 | Good. Number 14, Austin. |
| 00:37:48 | A over the- A to the five over B to the five. |
| 00:37:52 | Okay. And number 15, Rachelle. |
| 00:37:57 | A to the sixth and B- over B to the six. |
| 00:37:59 | Okay, and a rule. Cristie? |
| 00:38:02 | A M. |
| 00:38:04 | Okay. When we have the variable term in here, A divided by B, each term is raised to that power. |
| 00:38:15 | Okay. We cannot simplify because A and B are not the same terms. Okay. |
| 00:38:21 | Now, why I have you in groups, and this is what you're gonna do for today. And you're gonna present your answer tomorrow. |
| 00:38:29 | Knowing those rules of exponents... knowing those rules of exponents- |
| 00:38:36 | Knowing different properties, like distibu- distributive property, associative property, commutative property, |
| 00:38:42 | I want you to prove two things to me. |
| 00:38:44 | I want you to prove that A to the zero is equal to one. And I want you to prove that A to the negative N is equal to one over A to the N. |
| 00:38:58 | This is math homework for tonight? |
| 00:38:59 | This is what you're gonna work on in your group right now. |
| 00:39:02 | See if, knowing what you know about exponents, if you can come up with proof of this. Could you show me proof. Okay. |
| 00:39:13 | That's what you're gonna work on for the next 13 minutes. You'll present it to me tomorrow, in class. |
| 00:39:18 | What if we don't know? |
| 00:39:19 | Do we have to? |
| 00:39:20 | Are you serious? |
| 00:39:21 | Yes. That's why you're in groups. So start working on that. |
| 00:39:23 | All right. |
| 00:39:25 | Do you need me Robert? |
| 00:39:26 | If A equals one over- |
| 00:39:30 | I don't get that. A equals zero is another one, right? |
| 00:39:32 | Yeah. |
| 00:39:37 | So you're gonna keep plugging A in. |
| 00:39:38 | Oh, no, because lookit. A to the zero is A to the zero. So if you don't (inaudible). |
| 00:39:42 | If you plug in a number it's still A, 'cause there's no one. |
| 00:39:45 | Nuh-uh, 'cause if you plug in a number, then what if the A equals six? |
| 00:39:47 | Yes. See- yes, we're right. |
| 00:39:49 | Okay. |
| 00:39:51 | I can do A minus (inaudible), right? |
| 00:39:54 | Yes. |
| 00:39:55 | You- you have the question. |
| 00:39:56 | Okay, no. Okay, well- it- would this be- okay. |
| 00:39:59 | You know, zero times everything is zero, right? |
| 00:40:02 | Okay. |
| 00:40:03 | Okay. But, if A equals one then... then- One times zero is zero. Like... never mind. |
| 00:40:12 | You're not making much sense. |
| 00:40:15 | Never mind. Ryan made it sound better. |
| 00:40:16 | I always. |
| 00:40:18 | Okay. Well, just think about your rules of division and multiplication that you just did. |
| 00:40:24 | Work out some sample problems and see if you can come out and prove to me why A to the zero is one. |
| 00:40:30 | A- Because A is one. |
| 00:40:34 | No. A can be 10. And 10 to the zero power is one. A million to the zero power is one. Why is that? |
| 00:40:43 | Think about using those rules. |
| 00:40:46 | That like, just, went right over my head. |
| 00:40:51 | Yes? |
| 00:40:52 | Okay. |
| 00:40:53 | Okay guys. |
| 00:40:54 | He says that A zero plus N, equals one. |
| 00:40:59 | That's not right. |
| 00:41:00 | What? |
| 00:41:02 | Isn't that zero, not one? |
| 00:41:03 | That's A to the zero equals one. A to the zero power equals one. |
| 00:41:07 | Zero- A to the zero, but then it would still be zero. |
| 00:41:09 | Wouldn't it be zero? |
| 00:41:12 | Unless you can- |
| 00:41:13 | Because anything times zero is zero. Right? |
| 00:41:15 | But we have to prove to her why it's one. |
| 00:41:16 | Unless you have zero over one, and then you reciprocolated it. |
| 00:41:18 | It's not one, though. |
| 00:41:20 | Reciprocolated it? |
| 00:41:21 | That'll still be zero. |
| 00:41:23 | Okay. |
| 00:41:25 | I don't get how it could be one, though, 'cause anything times zero is... zero. |
| 00:41:28 | Is gonna be something that I can (inaudible) myself and (inaudible). |
| 00:41:31 | So then how could it be one? |
| 00:41:35 | Divide by zero. |
| 00:41:37 | You guys, you're trying to think of terms in- things in terms of numbers. |
| 00:41:41 | Think about putting your rules over here that we learned about, the adding and the- and the subtracting rules into- |
| 00:41:50 | If you took variables- don't do numbers- |
| 00:41:56 | M-A-B. |
| 00:41:58 | How could you prove something is equal to one? What do we know about the rule of one? |
| 00:42:04 | How are we gonna do these (inaudible). A by negative (inaudible). |
| 00:42:05 | Think about that. |
| 00:42:08 | Yes. You have a question? |
| 00:42:09 | We never learned these. The negative ones. Remember. We never learned these. |
| 00:42:13 | Oh, I know, but if you take the rules that you learned over here and think about that, you can make some examples. |
| 00:42:22 | Using either your multiplication or your division rules up there that you developed. |
| 00:42:27 | Let's see if you can come up with an example where that would be true. |
| 00:42:31 | Yes. |
| 00:42:32 | Hi. Since- Well, why do you have to multiply 'cause you didn't have to multiply it. Right? So it would just be one. |
| 00:42:38 | Really? What if A was a million? |
| 00:42:40 | Huh? A million? |
| 00:42:42 | Yeah. Let's say our A was equal to a million. |
| 00:42:43 | Oh, no. 'Cause it would be equal to six. Huh? |
| 00:42:44 | A million and one. |
| 00:42:47 | Equals six. |
| 00:42:48 | How do you- |
| 00:42:49 | And they- So six to the zero power is one. |
| 00:42:50 | Yeah. And any- |
| 00:42:52 | How? |
| 00:42:53 | 'Cause anything is one. |
| 00:42:54 | Well, that's what I want you to show me. How? |
| 00:42:55 | But if it says like- like when you multiply something by zero it says you're not doing anything to the- |
| 00:43:01 | Like if you had six times zero it says that you're not multiplying anything (inaudible) zero. |
| 00:43:04 | Okay. But we're dealing- we're dealing with powers now. |
| 00:43:07 | Yeah I know, but I don't- I don't see how it could make anything different. |
| 00:43:08 | So it'd be A times C. |
| 00:43:09 | Oh I know. 'Cause it's only one of these. |
| 00:43:12 | That's what I was trying to say. |
| 00:43:13 | Is that it? It's only one A because there (inaudible). |
| 00:43:14 | You're getting- you're on the right track. |
| 00:43:17 | 'Cause it's only one. It's not- |
| 00:43:18 | Start thinking about it in terms of multiplication and division now. |
| 00:43:20 | I know. I know. I know. I know. I know. |
| 00:43:21 | She's on the track. Yes? |
| 00:43:22 | I really don't get this. |
| 00:43:23 | Instead of- even if you- you were like- |
| 00:43:24 | Like, what- What if there was like one- One times zero, that would be zero, right? If it- if like (inaudible). |
| 00:43:29 | 'Cause if (inaudible) A were a number- |
| 00:43:30 | 'Cause if any number multiplied by zero is zero, right? |
| 00:43:32 | But is that what you're saying? That it's being multiplied? |
| 00:43:35 | It says A to the (inaudible) power. So- |
| 00:43:38 | So- |
| 00:43:39 | Does A to the zero mean A times zero? |
| 00:43:41 | No. |
| 00:43:42 | No it doesn't. What does it mean? |
| 00:43:43 | A would have to be one then, right? |
| 00:43:45 | No. 'Cause I can tell you that a million to the zero power is one. |
| 00:43:47 | Oh. So A times- A times itself to- |
| 00:43:50 | Ten thousand to the zero power is one. |
| 00:43:53 | What? |
| 00:43:54 | Over- Oh. |
| 00:43:56 | Start thinking about it. You're getting there, Brandon. Start thinking about it. |
| 00:44:02 | 'Cause these have to be A, they're A right? |
| 00:44:04 | 'Cause it'll be one over (inaudible). It'll just be (inaudible). |
| 00:44:08 | The first one we're having a hard time understanding. |
| 00:44:11 | Okay. |
| 00:44:14 | A to the zero power equals one. |
| 00:44:17 | So any number I plug in for A if I have it raised to the zero power, my answer's gonna be one. That's what I'm saying. |
| 00:44:24 | Now, think about what you can use over there between your multiplication and your division rules and prove to me that A to the zero is one. |
| 00:44:39 | Well, we thought we had it and then we weren't sure. |
| 00:44:41 | Well what did- what did you do? Let's see. |
| 00:44:43 | We thought that if A equaled one, then it would be true. 'Cause then- |
| 00:44:47 | Okay. And I'm telling you that if A is equal to 100, it's true. |
| 00:44:52 | (inaudible) |
| 00:44:53 | Yeah, but why? |
| 00:44:56 | How can I get something to the zero power? What types of things can we do? Using your rules up there? |
| 00:45:05 | Oh, okay. |
| 00:45:07 | Start thinking about that. How am I gonna get that zero power, and it'll start making sense to you. |
| 00:45:11 | Okay. Cause he kept saying this is impossible 'cause wouldn't it be zero equals A? |
| 00:45:15 | No. 'Cause what's two to the one power? |
| 00:45:18 | Two. |
| 00:45:19 | Two. |
| 00:45:20 | Okay. What's two to the second power? |
| 00:45:22 | Four. |
| 00:45:23 | Okay. And how did you do that? |
| 00:45:26 | I multiplied two by itself. |
| 00:45:28 | You multiplied it twice. Okay. |
| 00:45:32 | So here, this is not saying A times zero, this is say A to the zero power. |
| 00:45:36 | Told you. |
| 00:45:37 | A to the zero power. |
| 00:45:39 | That means you're not timesing it by anything so it stays A. So that would be one, right? |
| 00:45:43 | Well, not necessarily. What if A was equal to 1,000? |
| 00:45:47 | Then it'd be A to, I don't know. |
| 00:45:49 | No 'cause a thousand to the zero power is... |
| 00:45:52 | Is... |
| 00:45:53 | Is one. Now you have to prove that. |
| 00:45:56 | And think about it in terms of your multiplication and your division rules of exponents. |
| 00:46:02 | How am I gonna get A to the zero power as a solution. What can I do mathematically to get A to zero power? |
| 00:46:14 | Mathematically A to the zero power? |
| 00:46:16 | Mh-hm. |
| 00:46:17 | Oh you plus- plus N. |
| 00:46:18 | Can we use the calculator? |
| 00:46:19 | Yeah, you could use a calculator. |
| 00:46:21 | Okay. |
| 00:46:22 | You could use a calculator. |
| 00:46:23 | Okay. So- so the second- |
| 00:46:25 | Put in 100 to the zero power. See what they give. |
| 00:46:35 | Put in 1,000 to the zero power. |
| 00:46:39 | Oh, 'cause you're just timesing it by nothing, right? So it would be one, right? Or, you just get it. |
| 00:46:44 | How do you get zero equals one. |
| 00:46:46 | So no matter what number you're gonna put in to the zero power, you're gonna get one. |
| 00:46:50 | But using those laws right there, if you start thinking about it... |
| 00:46:56 | Work out a problem so that the solution is A to the zero and the light will go off. It will. |
| 00:47:03 | Start thinking, can I do a multiplication problem, can I do a division problem? Start working with exponents. Okay. |
| 00:47:10 | Okay, I don't get it. |
| 00:47:11 | Okay. Well- |
| 00:47:13 | Maybe you divide it by one or something, so that would be- |
| 00:47:15 | Well, how would you get a negative N up there looking at your rules of exponents? |
| 00:47:22 | Look at your rules of exponents up there, and where do you think you can get a negative N? |
| 00:47:26 | Oh, A times M over A times N equals A M minus N. |
| 00:47:32 | Okay. So therefore M would have to be larger than N. Right? |
| 00:47:35 | Yeah. |
| 00:47:37 | So you can prove that to me using that law. Think about it. |
| 00:47:41 | We don't understand this. |
| 00:47:43 | Okay. Now, Ryan, I figured you'd have this solved in no time. |
| 00:47:46 | I- I- I can't work with them. |
| 00:47:49 | Work with us? |
| 00:47:50 | It's weird. |
| 00:47:52 | We don't under... |
| 00:47:53 | I remember learning this a long time ago. Like, it was one of those questions (inaudible). |
| 00:47:57 | It was like one of those brain stumpers. |
| 00:47:58 | Yeah. And- |
| 00:47:59 | Where you had to like (inaudible). |
| 00:48:00 | Okay. So let's look at of our laws of exponents. We can add, we can multiply, and we subtract, and we can divide. Right? |
| 00:48:04 | And we can divide. |
| 00:48:07 | Yeah. |
| 00:48:08 | So think if I want you to set up a problem so that the solution would be A to the zero power. |
| 00:48:15 | What kind of problem could you give me? |
| 00:48:18 | I don't know. |
| 00:48:19 | A multiplication problem. |
| 00:48:21 | With exponents. |
| 00:48:22 | With exponents |
| 00:48:23 | With exponents. |
| 00:48:26 | A to the one. |
| 00:48:27 | Yeah. |
| 00:48:28 | Okay. A to the one and then what would I have to do to A to the one so that it becomes A to the zero. |
| 00:48:32 | Divide it. |
| 00:48:33 | Minus it. |
| 00:48:34 | (inaudible) divided by what? |
| 00:48:35 | One. |
| 00:48:36 | One. |
| 00:48:37 | By one or by- |
| 00:48:38 | Zero. |
| 00:48:39 | Let's see. Write that down. A to the one. Divided by- Okay. |
| 00:48:46 | Now since we're doing division, what does the rules of exponents say? |
| 00:48:49 | Oh you have to minus, right? |
| 00:48:52 | So it says that what you're going to do to get this, and we want this, put this equal A to the zero. |
| 00:48:58 | To equal? |
| 00:48:59 | A to the zero. Okay. |
| 00:49:00 | What do we do about division rules? |
| 00:49:05 | What do we know about division rules? What- what's- what's one of the rules? |
| 00:49:09 | You have to minus, minus. |
| 00:49:10 | You subtract the- |
| 00:49:11 | You subtract the top exponent from the bottom. |
| 00:49:13 | Yes. |
| 00:49:14 | So how am I going to- what am I gonna put down here such that this would be zero when I'm done? |
| 00:49:19 | One A? |
| 00:49:21 | A to the one. |
| 00:49:22 | A to the one. |
| 00:49:23 | Okay. |
| 00:49:25 | So it'd be A to the zero, and- |
| 00:49:27 | Now is A to the one over A to the one equal to A to the zero? |
| 00:49:31 | Yeah. |
| 00:49:32 | No. |
| 00:49:33 | Okay. Is it? By your rules, what do you do? |
| 00:49:35 | Yeah. You subtract. |
| 00:49:36 | 'Cause you subtract (inaudible). |
| 00:49:37 | You go one minus one is zero. |
| 00:49:39 | Okay. Now, what do you know about math when a number is over itself? |
| 00:49:44 | They cancel out. |
| 00:49:45 | And what does it equal? |
| 00:49:46 | Zero. |
| 00:49:48 | One. |
| 00:49:49 | One. One. |
| 00:49:50 | No. Come on. What does A divided by A equal? |
| 00:49:52 | One. |
| 00:49:54 | Okay. So, now if we do this division, what answer do we have here? |
| 00:49:57 | One. |
| 00:49:58 | One. |
| 00:49:59 | And what's that equal to? |
| 00:50:00 | Zero. |
| 00:50:02 | No. What does that say? |
| 00:50:03 | A to the zero. |
| 00:50:05 | And what is that equal to? |
| 00:50:06 | One. |
| 00:50:07 | One. |
| 00:50:08 | Now didn't you just prove it? |
| 00:50:09 | Yeah. |
| 00:50:10 | Yeah. |
| 00:50:11 | So how do you write that out? |
| 00:50:12 | Oh. I- |
| 00:50:13 | Can we just show that? |
| 00:50:15 | All right. |
| 00:50:16 | That's it. That's real easy. |
| 00:50:17 | Now, see you guys figured this out. So now with the next problem, we use the same type of logic to solve the second set. Okay. |
| 00:50:25 | All right. |
| 00:50:26 | Okay. |
| 00:50:27 | Now was it as hard as you thought? |
| 00:50:28 | Yep. |
| 00:50:29 | Sure. |
| 00:50:30 | Easy. |
| 00:50:32 | Okay. You guys, we need to wrap it up. |
| 00:50:35 | Shh! |
| 00:50:37 | We need to wrap it up. I will give you time in class tomorrow to finish up. |
| 00:50:42 | I know most of you will probably take this home and go, Mom and Dad help! Don't worry about it. |
| 00:50:48 | If you don't come up with a solution, we will work through and prove this for you. Okay? |
| 00:50:54 | Do we have homework tonight? |
| 00:50:56 | Yes, you have your assignments. You have homework tonight. |
| 00:50:58 | Oh. |
| 00:51:00 | See ya. |
| 00:51:01 | The first one. |
| 00:51:02 | Bye Mrs. Scott. I love you. |
| 00:51:05 | You're the best teacher in the whole wide world. We think you're the bomb. |