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.

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