# CZ4 EXPONENTS

This eighth grade mathematics lesson focuses on operations with numbers raised to certain powers, and calculating the area of a triangle and a trapezoid. It is the second lesson in a unit of work focused on raising numbers to powers. The lesson is 45 minutes in duration. There are 26 students in the class.

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00:00:19We didn't think we needed to bring it today.
00:00:21Absolutely not. Please, those of you that have not turned in the assignment yet, hang on to it.
00:00:37Under unusual circumstances, why don't you take it home, calculate the similar problems on your own.
00:00:48And my offer is... you all have a regular schedule on Friday.
00:00:54The information you read about not attending your music class is not accurate, that was an old schedule.
00:01:01So, you'll have all five classes and my offer or bonus is that those of you that want, I'm willing to help you at the sixth hour.
00:01:15Those of you that are having difficulty or don't understand the work can attend and I'm willing to devote some of my time for you.
00:01:24Considering the fact that in today's third class, you'll have the opportunity to write similar assignments as well.
00:01:34I'm offering you this opportunity, it's optional, okay? This Friday, after the last class.
00:01:41Today in our class, we'll calculate and review work which we discussed in our previous class. Turn to page 50.
00:02:05Before we start to go over this homework, we'll go over some of the rules which we did not discuss.
00:02:19Rules which we studied and calculated.
00:02:25We've dealt with powered numbers, which had concurrent elements and a powered number.
00:02:36If you recall, you had a homework assignment in which a grandchild was taken to a theme park and every...
00:02:45[Intercom]
00:02:54Is that supposed to be a class bell ring?
00:02:55I don't know. Every roller coaster ride cost him eight korun.
00:02:59Grandmother gave him enough for two rides, which is two times eight korun.
00:03:05Plus he received from his grandfather an additional five times eight korun.
00:03:12So he went on the roller coaster and he could have gone on the roller coaster how many times?
00:03:18Seven times.
00:03:19Seven times. Then we simplified it. Instead of letter A we used some other examples, some kind of gifts to your relatives.
00:03:30Do you recall that?
00:03:31No.
00:03:32You were saying or rather I encouraged you that you will pick two of your best friends and invite them to your Christmas dinner.
00:03:40So you'll buy each of them a gift. Two identical gifts, okay? You don't know what kind of gifts, but it will be two.
00:03:48But then you were at some kind of event and you said to yourself,
00:03:54I will expand the list of my close friends who will receive a gift from me. So, you will add five more gifts.
00:04:04See how it matches what's on top, which doesn't necessarily have to happen. So, how many gifts is he going to buy?
00:04:12Seven.
00:04:13Seven times D. Fine. And from that we derived a rule that we use addition and multiplication when the powered numbers are the same.
00:04:33Imagine instead of a gift, you'll give them a holiday card which will include a powered number.
00:04:42Give me a number.
00:04:46Eight cubed, fine. So you'll give them a card eight cubed, nicely wrapped. You'll give it to these two and to your five new friends.
00:04:59And now we can simplify it as...
00:05:04Seven times eight cubed.
00:05:06Seven times eight cubed. You can see that we work with the same powered numbers.
00:05:14The main number is the same as well as the exponent. Correct? Do you remember? Good.
00:05:24Then we had another category in which the only thing that was the same was, the main number or the exponent.
00:05:35If we look at it in the same order as we studied it. Anna?
00:05:39The main number.
00:05:40The main number is the same, fine. We learned how to deal with powered numbers with the same main number and we learned to calculate what?
00:05:54Open up your notebooks, the rules are in there. Open it up! And we're trying to figure out what, Kuba?
00:06:23We worked with powered numbers that had the same main numbers, Vasek? And we studied how to figure out the product and the quotient.
00:06:36And the quotient.
00:06:36Please tell me... we multiply two powered numbers with the same main numbers.
00:06:46Give me a calculation which includes two same main numbers. Ondras.
00:06:56Five to the sixth power.
00:06:57Five to the sixth power. I'll write it differently so you can understand the main concept of this example.
00:07:07It's the same, so five times five, which is the same. You've said to the sixth power and...
00:07:15To the fifth power.
00:07:16And to the fifth power. Please tell me guys, what remains. Jitka?
00:07:25Five to the eleventh power.
00:07:26Five to the eleventh power, fine. We've also studied... powered numbers with the same main number and its...
00:07:41Quotient.
00:07:42Quotient, good. Give me an example of two powered numbers that have the same main numbers. Vojta!
00:07:52Three to the eighth power.
00:07:54So, the two main numbers are three, fine, and we'll divide these two numbers.
00:08:01And in order for these numbers to be powered numbers we must include the exponents. Very good. Vojta!
00:08:08Three to the eighth power.
00:08:10To the eighth power and make sure this one is smaller.
00:08:12To the sixth power.
00:08:14To the sixth power, equals... is the main number going to be the same, Vojta?
00:08:18Yes, three squared.
00:08:20Three squared. Fine, that's another category. Here we worked with powered numbers that have the same main numbers.
00:08:31The next three rules apply to... Radka?
00:08:43Power of a product and power of a quotient?
00:08:46Yes. Tell me the power of a product, Radka?
00:09:00Two times three.
00:09:01Two times three, for example.
00:09:04To the fifth power.
00:09:06And the same exponent will apply to both of the main numbers. Because it will unfold into what... Andrea?
00:09:20Two to the fifth power.
00:09:21Two to the fifth power.
00:09:23Times three to the fifth power.
00:09:24Times three to the fifth power. You can see that the exponents are the same, do you see that?
00:09:31Here you have the same main number and the exponent, here just the main number and here we'll work with the same exponents.
00:09:41Furthermore, we know the power of what, Marketa?
00:09:47Quotient
00:09:48Quotient. Give me a simple example.
00:09:51Eight divided by two to the third power.
00:09:57To the third power, fine. And even here, we'll have the same exponent. Its product.
00:10:08So, it's going to be eight to the third power...
00:10:12Divided by two.
00:10:15I said product, excuse me I meant quotient. Okay, and we also studied what else? Katka?
00:10:26Power of the exponent.
00:10:27Power of the exponent. Here we go. Give me an example.
00:10:33Two cube and the entire thing is to the fifth power.
00:10:39To the fifth power.
00:10:42This equals to? Lucka Maresova?
00:10:47Two and we multiply it by...
00:10:50This number stays the same and the exponent will be...
00:10:55We multiply three times five.
00:10:57We multiply it. That is three times five equals 15. It's two to the fifteenth power. These are rules which you must master.
00:11:14Those of you that are having some difficulties with this, should practice at home with his friends in order to get comfortable with this.
00:11:26Okay, let's look up page 50.
00:11:31Open up your notebooks and write today's date, twenty-eighth of November and we'll do problem three, five letter A.
00:11:41One problem after another, this time we'll do the work together with someone up here by the board.
00:11:47First student to come up to the board is Paul. We must... as I told you yesterday, we must write down the assigned problems.
00:11:59Okay, Paul. Calculate 30 to the third power and describe to us how you're going to figure it out.
00:12:12First I multiply three to the third power.
00:12:15You'll raise the powers.
00:12:15Raise the powers of three cubed.
00:12:17Which totals?
00:12:19Seven is 27.
00:12:20Wait a minute. Don't write three cube, that would confuse all of us.
00:12:26Seven equals to 27.
00:12:28Correct.
00:12:30Plus-
00:12:31No, not plus. Just write the number. It's actually...
00:12:40Twenty-seven and three... three to the tenth power is 1,000 so that is 27,000.
00:12:47Excellent. The answer is 27,000. Call up the next person and read them a problem.
00:12:52Alena!
00:13:00No one is absent today?
00:13:01Martusova is absent.
00:13:03Negative three to the fourth power, you're supposed to compare it.
00:13:12Okay!
00:13:13And negative five to the third power.
00:13:15Leave a blank between the two.
00:13:16Okay!
00:13:18Lucka was absent in the first class as well, correct?
00:13:25What is the exponent?
00:13:27To the third power.
00:13:29So, it will be... negative three to the fourth power is larger because we have an even number, negative coefficiency.
00:13:41Yes, the exponent is an even number but in the other one, the exponent is an odd number so what kind of number will you come up with?
00:13:47Negative.
00:13:47Negative, but in this one the number will be...
00:13:49Positive.
00:13:50Positive, fine. So do you have it correct or not.
00:13:56I have it correct.
00:13:57Incorrect, you need to reverse the numbers.
00:13:59You described it correctly but you wrote it wrong. Good, next person, let's hurry up.
00:14:15Write the main number two, 128.
00:14:33Number two is the main number so write number two and now find out the exponent.
00:14:38If it's two squared then it's four, so try to the third power which is eight and so on. Who's finished with this problem?
00:15:03Continue by multiplying it here on the side. Two, four. Here try this marker.
00:15:48Who can help him out. Michael Onderzak.
00:15:56To the seventh power.
00:15:59Two to the seventh power.
00:16:02(inaudible)
00:16:03That's fine. Two to the sixth power is 64, two to the seventh power is 128. Excellent.
00:16:09You know how you came up with this, yes? Let me have the marker.
00:16:11Two to the first power equals two, squared is four, cubed is eight, times two is 16, which is the fourth power.
00:16:21Thirty-two is to the fifth power, 64 and 128. Call up the next student.
00:16:30(inaudible)
00:16:37We're not done yet.
00:16:41Seven squared times seven cubed. Write the exponent of seven.
00:16:48That equals to seven and I multiply two times three.
00:16:53No, no.
00:16:55I meant adding it up, so the answer is seven to the fifth power.
00:16:59Show us which rule you followed to complete this problem.
00:17:02That one up there.
00:17:04Okay, why don't you pull down the board and point it out to us. Very good. Next person!
00:17:11Anna! Six times three cubed, minus four times three cubed.
00:17:37Six times,
00:17:40Don't put it in parenthesis, six times...
00:17:43Three cubed.
00:17:45Times, she said times so it's multiplication. Three cubed.
00:17:53Minus four times three cubed.
00:18:00Before you start calculating it, which rule will you use in this case.
00:18:08It's an assignment which has the same main number as well as the exponent.
00:18:14Power of the exponent.
00:18:16Take a good look at it. Show me the two powered numbers in this example. Show me the first one? Read it out?
00:18:26Three cubed.
00:18:28Three cubed.
00:18:29Which rule applies to this?
00:18:38That one.
00:18:39Correct. These are formulas that have the same main number as well as the exponent.
00:18:46Okay, six times (inaudible) minus four times (inaudible) is the same as six times three cubed. Yes of course. Which equals...
00:18:57Three cubed is 27 times two equals 54.
00:19:05Do you all agree with the answer?
00:19:06Yes!
00:19:07Katka, do you agree?
00:19:08Yes.
00:19:08Correct. Is there a simpler way of doing this? You may sit down now.
00:19:14Isn't there a simpler way? Two times three is six and calculate six cube. Why not.
00:19:21(inaudible) It's not possible.
00:19:23We would not have to go through all these steps. Jitka!
00:19:28(inaudible)
00:19:29Why can't I do two times three is six cube. Ondra?
00:19:38Because, power of a product surpasses everything else.
00:19:40Because, raising powers surpasses multiplication, correct. Anna did it correctly, except she did not call up the next person.
00:19:49Yes, Lukina!
00:19:50Come up here Vasnovsko. Catch Lukina.
00:19:59One point two to the eleventh power divided by one point two to the ninth power.
00:20:09Tell him the instructions on what to do with the problem.
00:20:12And you're supposed to calculate it.
00:20:14So, which rule will you follow? Stand aside and tell us which rule you will follow.
00:20:21This one.
00:20:24We're dealing with power of quotient with the same...
00:20:29Same main number.
00:20:29Very good. Equals! You're supposed to calculate the problem, you just simplified it.
00:20:40Calculating the problem means, you come up with a number. You can calculate it in your head.
00:20:48First, you calculate 12 squared, which is how much? Just ignore the numbers that are to the nearest tenths for now.
00:21:03That's 144, and you separated the two because of the decimal numbers (inaudible). Very good.
00:21:12You just didn't underline the answer twice. Next person.
00:21:32Eight to the X power divide it by eight to the third power equals eight to the eighth power. (inaudible)
00:21:45So write next to it X equals...
00:21:48X equals.
00:21:50Point out the rule you're going to follow! Correct.
00:22:06X equals eight...
00:22:12No, no. Our only concern is this X.
00:22:14X equals...
00:22:16We can't be mixing it up, okay?
00:22:18X equals 11.
00:22:20Correct, how did you figure that out? How did she come up with that answer, that X equals 11!
00:22:25I added up number three and number eight.
00:22:27Fine and in order to verify it, try to insert the number in there. Instead of X, write in number 11.
00:22:35Eight to the eleventh power divided by eight to the third power equals to eight to the eighth power.
00:22:44Excellent. Please underline X equals 11. Underline X equals to 11, that's the answer. Good, next person.
00:23:04Andulka!
00:23:07Write down a number which is greater. Seven squared in parenthesis. (inaudible)
00:23:10May I write it like this. Is it Okay?
00:23:14Here you did not write what X equals. Please write it in.
00:23:18And seven to the third power.
00:23:22That's why we're doing this together in class.
00:23:29First of all I calculate what's in the parenthesis. Seven squared equals 49.
00:23:36Is that important? Which rule will you follow for this problem. Move aside and point it out for us.
00:23:43That one?
00:23:49Correct, so use it!
00:23:54Here we have two X and three times three is nine.
00:24:01Yes, very good. Why don't you come over here to write on this side of the board.
00:24:34Equals, good.
00:24:37Three times 10 to the fifth power.
00:24:42He'll figure it out on his own.
00:24:44Plus five times 10 cubed.
00:24:50Very good Michael.
00:24:52Plus two times 10 to the first power, plus six times one.
00:25:08Very good, call up the next person. It's correct. I'll just ask you. Could this have been just two times 10, not 10 to the first power?
00:25:16Yes.
00:25:16Because 10 to the first power is 10. The way he did it is correct as well because that's how we studied it.
00:25:24Why is there so much noise in here? Here he wrote times one. Is that correct?
00:25:30Yes.
00:25:30Absolutely it's correct. He also could have... instead of one, he could have put in times...
00:25:34Six times...
00:25:37Six times...
00:25:38Ten to the N power.
00:25:39Ten to the N power. Both answers are correct.
00:25:43Write A times 10 to the N power.
00:25:48Write the formula, A times 10 to the N power. That's what we can use when we deal with large numbers on a calculator.
00:26:01Seven hundred and thirty-four thousand...
00:26:20No, no.
00:26:23Leave it up there. In order for you to have 10 cubed here and not be a mistake as the final answer.
00:26:32You would have to have over here...
00:26:34Seven hundred and thirty-four.
00:26:34Seven hundred and thirty-four. That's still not a final answer. This will equal...
00:26:51Very good.
00:26:52No, that's not correct. It should be times 10 to the fifth power.
00:27:05Underline the answer twice. Good. Ladies and gentlemen, look on page 51, exercise number one.
00:27:19Please take out a number two pencil, number two pencil and you're supposed to calculate exercise number one.
00:27:36Write it into your notebooks as well.
00:27:38Using the pencil?
00:27:40Excuse me?
00:27:40With the regular pencil?
00:27:42Start with the outline of 51 over one! I will dictate you the following assignment. Negative 50 in parentheses to the fifth power.
00:28:00I thought we were going to complete the assignment in the book but instead we'll copy the assignment into our notebooks.
00:28:04I'll start over again. Negative 50 in parentheses to the fifth power plus negative 30 in parentheses to the third power.
00:28:14Plus negative 20 in parentheses to the second power, plus negative 10 to the first power, plus zero to the tenth power,
00:28:28Plus 10 to the first power, plus 20 squared, plus 30 cubed, plus 50 to the fifth power.
00:28:42And I would like to find out if you can figure out why did I ask you to use a regular pencil. Figure this problem from your memory.
00:28:58How is the pencil going to be useful to you. Marta!
00:29:04So we can reduce a fraction... to the lowest term...
00:29:08Not to reduce a fraction but to calculate...
00:29:11When we have negative 50 to the fifth power plus 50 to the fifth power, it will equal to zero. (inaudible)
00:29:16Yes of course. So you cancel out those numbers that equal to zero. For instance, negative 50 to the fiftieth power and plus 50...
00:29:32Excuse me, I just made a mistake. Negative 50 to the fifth power and plus, Tom?
00:29:37Plus 50 to the fifth power.
00:29:39Fifty to the fifth power, very good. Just cross it out lightly.
00:29:44This does not mean you're eliminating these numbers, it just means you're done with those numbers.
00:29:54Can you find another pair of numbers which can be crossed out? Jitka!
00:30:00Negative 30 to the third power plus 30 to the third power.
00:30:02Very good, those two equal to zero as well. What else? Kuba.
00:30:10Negative 20 squared and 20 squared. I see.
00:30:14Does anyone object to that? Tom objects, who else.
00:30:18Me!
00:30:19Why not... why do you object to that answer. Kuba.
00:30:24Why didn't some of you have a problem with negative 30 cubed but object to negative 20 squared?
00:30:35Negative 30 cubed was fine but you object to negative 20 squared. Do you know, Ondra? Please tell us.
00:30:54There is an even number... it's an even exponent.
00:30:59The negative main number has an even exponent. That's not the same thing, is it! Iveta.
00:31:07Because a positive number will come out of this.
00:31:09Because in this case we get a positive number. Negative 20 squared does not come out to a negative answer but a... Kuba!
00:31:15Positive number.
00:31:16Positive number, so these two numbers can't be crossed out. What else can we cross out... which other pairs equal zero, Vojta?
00:31:26Negative 10 to the first power.
00:31:30Yes and...
00:31:31And negative... and positive 10 to the first power.
00:31:33Good, do we all agree?
00:31:35Yes!
00:31:37Are there any other pairs we can cross out? Vasek.
00:31:44No, maybe zero?
00:31:48Okay, let's try that one, why not. Zero to the tenth power equals...
00:31:52Zero.
00:31:52Zero, so what numbers remained untouched and remain to be calculated. Michael!
00:32:07Negative 20 squared and plus 20 squared.
00:32:13What's the result?
00:32:15Eight hundred.
00:32:16Eight hundred, write in 800.
00:32:19Negative 20 squared equals 400, plus 20 squared is also 400, and that totals 800.
00:32:27Wow, we're moving along just fine. At home you will verify number two, my original idea was to do it in class.
00:32:41You'll do number three and four large A and small A will do. Alenka up here on the board.
00:32:55Number four A and small A. You need to calculate the second root of a number which is in the parentheses.
00:33:07First of all, figure out the estimate, so the second root of six, second root of six.
00:33:15(inaudible)
00:33:17Next math class which is tomorrow.
00:33:20(inaudible)
00:33:22Oh yes, that's correct. So it's for Friday. Quiet. First of all you must figure out the estimate.
00:33:32Estimate.
00:33:34The second root of a problem, we do an estimate based on what?
00:33:37Based on the closest squared number.
00:33:39Correct.
00:33:40The squared number is four and (inaudible).
00:33:49Fine and now do the calculations. You're supposed to do it to the nearest decimal number. What do you need for that?
00:34:04Table chart.
00:34:04Go get it. Exactly to the nearest decimal number. Very good. Can someone calculate it on a calculator?
00:34:38Two point four, four, nine, four, eight, nine, seven.
00:34:40So based on the calculator the number will be rounded to... Alenka, come here.
00:34:46This on the board is correct, she followed the table chart.
00:34:50What is the number again?
00:34:51Two point four, four, nine...
00:34:58We don't need to go that far. That's enough. Equals to approximately... just write it in. It's more accurate with the calculator.
00:35:16Because, the table chart is rounded to the nearest tenth, okay? This nine here was shown in the charts as 545.
00:35:30And you continued to rounded up but there isn't much of a difference. Call up the next person, Alenka!
00:35:58The second root of 60, the second root of 60.
00:36:05Rounded up to the nearest decimal point. First of all, do the estimate. That will be an approximate second root of what nearest squared number?
00:36:16Sixty-four.
00:36:19Wait a minute. Not from 64 but from the root of 64 and that equals?
00:36:28Very good, that's your estimate. Now calculate it. The second root of 60 equals... use the graph chart.
00:36:57And now round it up to the nearest tenth.
00:37:01[Laughter]
00:37:06It's a good thing the gardeners didn't put water in it. Thank goodness for that. To the nearest tenth is eight. It is accurate, fine.
00:37:27Students, please complete A, five large A, six and seven large A...
00:37:37We're not going to do it in class?
00:37:38Well yes but it's taking us too long to complete these problems. And on page 53... I thought we would have time to do this here in class.
00:37:48On your own or in a group of two, please do number 16 right now, okay?
00:37:56Now?
00:37:57Yes, now.
00:38:02The entire page of 51.
00:38:06Please do the problem in your personal notebook, okay? I'll be filling this in.
00:38:36You may work in a group of two, if your group is still having troubles you may turn to another group for assistance.
00:39:17You will work with an area of what kind of geometric configuration?
00:39:21Trapezoid and triangle.
00:39:22Triangle and...
00:39:24Trapezoid.
00:39:25Trapezoid. To calculate the area of a triangle, you need to know what.
00:39:29Height and length.
00:39:30The length and height of one side. To calculate trapezoid, you will need to know...
00:39:39A and C.
00:39:41So, the base and the height, very good. And you'll calculate the height based on your knowledge of the Pythagorean theorem, which are marked in here.
00:40:04Everyone who completes the problems will get an A grade. A very high A grade. We have three minutes remaining in class.
00:41:32Use what's on the board... oh you mean that one, yes you may use a calculator.
00:42:27Don't forget that these problems require that you reserve about 15 percent, okay?
00:43:27Don't spend too much time on this. You have this here... hurry up so you can finish it up. You can just calculate the triangle here.
00:44:08What is this? Height?
00:44:12The height of this triangle.
00:44:15Of this triangle and this is the height of...
00:44:19Trapezoid.
00:44:20Trapezoid, correct. Do you need an additional minute to complete this... do you need it?
00:44:47Yes, yes.
00:44:48Fine, I'll give you one more minute. Actually, you'll get two minutes.
00:47:02Excuse me?
00:47:03How much...
00:47:03The answer is obvious... well it is a verbal answer. Let me see.
00:47:18Two hundred and one point twenty.
00:47:21Why don't you round off the number.
00:47:22So, it will be 200.
00:47:26Lukine, would you be able do describe the entire process?
00:47:29What do you mean.
00:47:29The process...
00:47:30Well, some of it.
00:47:32Well, he helped me on something and I helped him with something else.
00:47:35Okay, both of you can write an A grade in your notebooks. Congratulations.
00:47:45Lukine, you will come back here with Tom later for me to sign your grade. Let me see what you got. No, no, no.