 So exponential and logarithmic we're going to introduce an entirely new function an entirely new button on your calculator By the way, you want to be thinking about if you haven't if you're whether you're going to buy a graphing calculator or rent one from me You're planning on borrowing one from the school. You need to bring a hundred twenty five dollar deposit check in the next couple of days Or you can purchase one from staples for probably about a hundred and ten bucks. I would borrow one anyways We're going to be introducing an entirely new button on your graphing calculator on your any calculator It's gonna be the log button, but that comes later review exponent laws The parts of a power are listed below. So we call this entire thing a power We call this thing that's at the base of the power at the base. That's a convenient name We call this an exponent and in grade 9 and grade 10 you learn the exponent laws and Believe it or not. Yes You go get your book hustle, please And believe it or not as far as I know, this is it for exponent laws. I Have a math degree. I didn't learn anyone. I learned how to apply them in cleverly. I could modify some of these Yeah, let's bring them from now on folks So the first one is the product law The product lot x to the m times x to the n for example Had my way. We'd all leave right now and go to a different classroom Need to find someone who's got a block a prep X to the m times x to the n when you had don't write this down when you had something like this x to the fourth Times x to the fifth. What was the answer? Louder So what do you do with the exponents when you're multiplying in the bases are the same? So we would write this as x to the m plus m Now that you've seen how we're going about this What was the rule when you were dividing exponents and the bases were the same? What did you do with the exponents x to the m minus m? What was the rule when you had a power to a power when you had an exponent on an exponent? Do you remember? Don't you multiply so I'm gonna write that as x to the m and because if I don't write anything that is multiply What about if you had several things in brackets a product being multiplied together What did the exponent do? I'll tell you this one because you all know what you don't quite realize what they're asking for What we wanted you to realize was the exponent would go on to the first term and on to the second term Go on to everything and if there's a third to what as long as there wasn't a plus sign in between them If there was a plus sign then you had to foil with brackets and all that stuff. That's a different entirely different rule Power of a quotient This is also the same idea what we want you to realize is that if you have a fraction all to an exponent That's really the same as Changing mr. Do it That's really the same as the top term to the exponent Divided by the bottom term to the exponent Oh, but we're being fussy the denominator can't be zero why because you can't divide by zero Why I think I showed you my little six divided by two six divided by excellent stick when we did reciprocal Then we have this you did most of this in grade nine, I believe Then we have the integral exponent rule. I think you do this in grade nine, but you do it more in grade 10 What did a negative exponent mean? How could I write this as a positive exponent? One over the X back. Here's how I teach my grade 10s I always tell them that a negative exponent acts like an elevator and causes that term to change levels So this would be Different color mr. Do it one over X to the M you see my rule is more general because if I gave you this I Would say a negative exponent acts like an elevator and it causes it to change levels. It's in the basement Now it's on top. In fact, you're going to hear me frequently say during this lesson Elevator what I'm really saying is move this stuff up and down and then the very last one you learned in grade 10 Is what a fractional exponent means for example, don't write this down, but something like this eight To the two-thirds. I'll give you a hint It involved that symbol suddenly showing up and the eight ended up there It gave you certain roots. Yeah, you remember I'll give you hits either the square root of eight cubed or the cube root of eight squared. Which one say it louder. You're right So Brett said it's the cube root of eight squared which it is now Really kids kind of remember this the real issue is does the two go there or there? Does the three go there or there allow me to show you what my grade 10 teacher taught me all these years ago? She taught me flower power Say what she said when you have that fraction The top is the flower and the bottom is the root of the flower and the root goes to the root and the flower Goes to the power and so she always gave us the little thing called flower power for us to remember What goes where you can use whatever you want to? But you got to keep straight what goes with what so in our generic rule down here This would end up being flower to the power root to the root and then it says or What we want to write here is this to the M What it's saying here Trevor is look If it's easier to do the root first and then do the exponent go ahead If it's easier to do the exponent first and then do the root go ahead the order here doesn't matter Pick whichever makes the math easier It's something that really depends on the numbers that are there those are your exponent rules. That's it Let's use them. This says write each expression without brackets and with positive exponents Here I have two things being multiplied together. I have a one-half and I have a why The one-half is a fraction. Is this a fraction? Yes, it is because it's over. What it's invisible, but it's over what See, here's how I would handle this then I would say to myself self on the top I have a one on the bottom. I have a two and this thing here is a negative exponent put it on the elevator I don't let myself get confused with how do I multiply fractions? I don't need to one term at a time What's on top? What's on the bottom? now There is an order of things that I do and this is mr. Dewey's method. I have not yet seen this in any textbook I tried to suggest this to some math teachers because by the way, I think it's an easy one I can do almost all these in my head. I'm gonna teach you how to do that technically My method violates bed mass technically But it will always work anyways It's one of those times when the order of operations you can do it in the wrong order and still get the same answer and by doing so It's way way easier. So I'm gonna say try my method up here. We have the exponent rules. I always do These two first always I'm always looking for a power to a power. Is there a power to a power? Is there a power to a power? That's my checklist. I always do this next I Always get rid of any negative exponents and put them on the elevator. I cause them to change level By doing that, I end up with something that I can almost always do the rest of it in my head in one step by cheating cleverly so When I go to be I say are there any power to a power are there any brackets with exponents outside of them say no Say no So I did that check though. Now elevator Does the five have a negative exponent on it? Leave it Does the X have a negative exponent on it? elevator I've taken care of the top. Does the Y have a negative exponent on it? Elevator Have I taken care of every term good? I'm done see I See brackets. Are there any exponents outside of the brackets? No, I'm going to do elevator first always and this is where you could argue that I'm starting to violate some bed mass rule But I want to show you why this makes it so much easier what I'm trying to get rid of here Haley is the Negative exponent divided by a negative exponent where you're supposed to go a minus minus and what they I'm gonna ditch that totally watch I'm gonna say I like the four. I like the X cubed. I like the Y. I Like the two X to the fourth ends up down there. I like the Y squared You okay with that Haley? Now here's why that makes life so much easier that one step I will almost always write out and I can now do the rest in my head first of all Final answer numbers by numbers. What's my coefficient going to be? Can you see it? See the eight how many X's on top grand total? Three how many X's on the bottom grand total for how many X's left and where? Did you hear me ever say negative in there and that we Bypass the whole negative exponent at what the heck do I do we said look I know there's gonna be a single X on the bottom How many Y's on top grand total? Three see him see him How many Y's on the bottom grand total trick question? None how many Y's left and where three on the top? That's how I do these I always get rid of negative exponents as soon as I can And then I just say how many on top how many the bottom how many left and where and Trevor it gets rid of the Whole do I do a minus minus here? Do I do what if I have a negative exponent on the bottom? I never will Never will Why not make my life easy? Also, that's where I'm gonna make my most stupid mistakes. So why not get rid of that step? Gotta be full disclosure though technically Somewhere along the way if you're paying attention, we violated bed mass. Don't worry about it. It'll always work. Yes To get the X Okay, so we're looking here. How many X's on top grand total there's three of them See him how many X's on the bottom grand total four of them when I cancel how many will be left and where? One left on the bottom Okay, now the see the other way you could have done this as you would have gone Well, these are being multiplied. So I add the exponents. I have an X of negative one What the heck do I do with the X and negative one? Oh, I better put that as a frat I managed to avoid all that and do it in one step, which I think is a better method Trust me D. I don't know how your math 10 teacher taught you but I'm gonna guess that your math 10 teacher might have told you to go Well, these are being divided right here. So you're gonna go negative two negative three minus forget that I got no use for confusing myself. First of all, are there brackets with exponents on the outside? No, good elevator Let's change colors. Mr. Do it The 24 I'm happy with the M to the fifth. I'm happy with I do not like that P. I'm gonna put it down there elevator I'm good with the Q to the fourth I'm good with the negative four. Wait a minute. Isn't that an elevator? It's not a negative exponent Ryan That's just a negative number right negative exponent moves levels a negative number stays where it is I'm good with this M to the fourth I'm good with this P squared But this Q of the negative two elevator Move to the top by doing that. I can go to my final simplified answer now in one step ready Ellen Numbers first. I got a 24 on top. I got a negative four on the bottom. What is 24 divided by negative four on top? How many m's on top grand total? Five how many m's on the bottom grand total for how many m's left and where? How many Q's on top grand total six how many Q's on the bottom grand total none How many left and where six on top? Which letter haven't I tackled yet? Have you kept track? Yeah, okay, it's easy to keep track. How many P's on top grand total none How many on the bottom grand total five how many left and where five left on the bottom and That avoids the whole minus minus dividing by a negative what that gets rid of it and that's simplified E is there an exponent outside the brackets in E? Then we do that first and I remind myself as our that it goes on to everything I usually draw those in because it takes all of one second it's going to be Three to the third X to the to the what? six Y to the what? nine I Would accept this They might write three to the third is 27 what's the rule for that honestly? The rule is if you can do it in your head write it as a number if you can't do it in your head don't write it as a number 27 X to the 6th y to the 9th is there an exponent outside the brackets Elevator I'm happy with the 12. It doesn't have a negative exponent. I don't like that be the negative a half I'm going to move it to the denominator I've got a three down here. I'm good with that and I've got a B down here. I'm good with that Numbers first I got a 12 on top. I got a three on the bottom. What's 12 divided by three? Okay, by the way, don't write this down Carly. Just watch what if this question instead? Had the three on top and the 12 on the bottom I'd still have a four But where would it be on the bottom? Okay, so in other words, you're kind of reducing fraction You don't keep stuff in its proper place Jumped suddenly So if I simplify this there's going to be a four How many bees on top? none How many bees on the bottom grand total? One and a half. Can you write that as an improper fraction because one and a half? I don't write this down this Is bad math matters because also you can't tell if that's one and a half or one to the one half power It's really awkward to tell so what is one and a half as an improper fraction? Three over two so you could do that now. I don't know what answer they're looking for We could also because I have a fractional exponent. We could flower power it We could go for all over the root symbol The bee goes there root to the root flower to the power Now technically you don't need to put the two there to mean square root We decided a long time ago as math nerds that because square root is the most common root if you leave it blank We assume it's a square root. We assume there's a two there That's why you never bothered writing a two there, but they're technically is turn the page says find the exact value of the following and verify with a calculator Okay Nine to the negative three flower power One over nine cubed no calculator What's nine to the third? Correct me if I'm wrong, but if you turn back one page to the previous page is the facing page a blank page Or not Yes page what page number is that? Turn to page 90 write this down on page 90 Exponents worth Memorizing this test the log test the first half of it will be non calculator I'm going to expect you to memorize certain exponents Well, I'm fibbing Memorize isn't quite the right word Tyler I'm going to expect you to recognize certain numbers and when you see them you should know hey That's two to something or three to something. Here's what you need to have memorized write these down, please two squared Two cubed To the fourth To the fifth two to the six with no calculator What is two squared? Four what's two cubed don't you dare say six eight? What's two to the fourth? I'll give you a hint. It's this times two sixteen What's two to the fifth? I'll give you a hint. It's this times two thirty two Two to the six to sixty four What do I mean by you need to have these memory memorized? Mitsu when you see a 32 you better right away say I know that's two to some power and Then I can figure it out two four eight sixteen thirty two sixty four a hundred twenty eight hundred two six five hundred and twelve one thousand twenty four two thousand forty eight four thousand ninety six eight thousand one hundred ninety two sixteen I can keep going if I need to They're a bit more familiar to you guys because they're also computer numbers because computers are all powers of two You may have noticed that any RAM chip is actually a power of two You need to know this Three squared three cubed three to the fourth three to fifth. What is three squared nine? What is three cubed? 27 what's three to the fourth? I'll give you a hint It's actually three squared squared. It's nine times nine What it is 81 in your head multiply 81 times three to get three to the fifth And we all get two hundred and forty three don't we? Eight times three is twenty four one times three is three twenty four three You need to know this Four squared four cubed and sort of four to the fourth it pops up once in a while Four squared is sixteen Four cubed we all know is sixty four Four to the fourth is sixty four times four You need to know this and for the rest of them really all you need to know are the squareds and the cubed So we're gonna fill up the rest up to ten five squared 25 five cubed it's 25 times five boys and girls 125 you need to know six squared and six cubed Six squared is 36. What's six cubed? Well, it's gonna be 36 times six 216 and again Brienne when I say you need to know what I'm really saying is from now on for the rest of the year when you see the number 216 right away you should be thinking hey, that's an exponent of something and I'll give you a hint if they end in six They're almost always an exponent of six except for 16, but every power of six ends in the six Don't believe me play with your calculator. You need to know seven squared and seven cube Seven squared 49 seven cubed Look up. It's not 49 times 7 if you're doing it in your head It's 50 times 7 minus 7 because 50 is a much nicer number to do math with. What's 50 times 7? 350 minus 7 because it's really 49 times 7 343 I Get 243 and 343 mixed up You know why because they both had in 43 And they're both from prime exponents and I can't fly anyways I've just given you a way though it if you ever need to find 7 to the third. It's 49. No 50 times 7 minus 7 You need to know 8 squared we know 64 Also happens to be 2 to the 6th power Also happens to be 4 to the third power 8 cubed 64 times 3 I'll be honest this one you don't really need to know because I don't have it here This is one of the ones that when I see it I recognize it and I go that's 8 to some power So somebody on your calculator was a what is 8 to the third power? 512 that's right 512 Which is also a power of 2? Why is anything that has an 8 in it a power of 2 as well because 8 and you need to know 9 squared. Oh, mr. Duke. You've been alternating colors. Why not continue 9 squared and 9 cubed 9 squared is 81 9 cubed which brings us to the question that we were just doing I think by the way 9 cubed is Look up 80 times 9 plus 1 more 9. What would 80 times 9 be? 720 plus 1 more 9 again It's worth memorizing them. Well hang on Carly once you're done yawning when we get to the test if you don't have them memorized you better write them out Most of you will just memorize them because we're gonna be using all of these so often during this unit And you're gonna get tired of having to look them up or go to your calculator your natural teenage lazy streak We'll take over and your brain will say look memorize these stupid things because I'm tired. Look at these up So it will probably take care of itself back to the notes So I'm guessing mr. Duke. This is one over 729 why yes it is flower power The what root of what to the power of what? the fourth root of 81 To the third I said to you the order that you do this doesn't matter You could go 81 cube first and then take the fourth root of that Do you know 81 cubed in your head then you know what let's do the fourth root first What's the fourth root of 81? Some number to the fourth power equals 81 Give you a hint turn back a page 81 appears twice in one of those columns it appears next to something to the fourth power, huh? This is actually the same as three to the third. Oh and three to the third. I know is don't say nine See 25 to the zero What's anything to the zero one? Why I have a lovely explanation, but another time D. You know what I'm gonna do first in D Elevator this is one over 16 to the one half Now This is flower power, but this is a shortcut. That's just worth remembering one half as an exponent is a very special route What is one half as an exponent? What route is one half as an exponent? Yeah, this is one over the square root of 16. I'm we're gonna write this in our notes This is one over the square root of 16 But if you jumped straight to the final answer of one over four without writing this intermediate step I'd be happier There are two exponents that are fractions that are worth memorizing a one half power Which is square root and what was a one third power a one over three power? Cube root. We're gonna use those pretty interchangeably. Whatever we want skip this Do this Here is where my method that I've showed you Will start to make you a wee bit happier. You'll say oh Derek you should publish this I Tried bringing it up at a math meeting but because it technically violates bad math. They refused Okay So I'll just continue to do exponents. I hadn't get all the right answers with less work What was the first thing that I said we look for is there an exponent outside the brackets? Yes, there is that exponents gonna go on Everything it's gonna be Four to the negative three x to the negative three y to the positive six The next thing I told you to do was elevator always get rid of negative exponents This four is gonna move to the bottom this x to the third is gonna move to the bottom This y to the six is done is four to the third one of the ones that I told you to memorize Then I'd expect you to actually not write four to the third. What is four to the third power? I would probably if I gave you this as a multiple choice This is what I would have is the answer and this is what I would prefer you to write B. Ah, don't panic First thing you'll look for are there brackets with exponents outside the brackets? Yeah, take care of those first negative a b to the negative six All over b to the negative one a to the eight Right power to a power rule Sorry, yeah, I'm right I missed a fifth I missed a five Let's try that it there. Don't forget your exponents Now what elevator Is this a negative exponent? No It's a negative number. In fact, what number is sitting right there? It's invisible Yeah, I'm going to just treat that as a negative one kicking around, but I'm not going to write it I'm just going to remember it. I'll drop the negative down I'm okay with that a to the fifth because it's not a negative exponent b moves down there What about this b here elevator moves to the top What about this a stays And now we can go to our final answer. First of all numbers. What number is on top sitting there a negative one? What's on the bottom a one what's negative one divided by one? There's going to be a negative kicking around How many a's on top grand total? Brett how many a's on top grand total? How many on the bottom grand total? How many left and where? See, I'm trying to avoid the whole having to go five take away eight get an a the negative three and then freak out because It's like we don't need to There's three left on the bottom. How many b's on top? How many on the bottom? How many left and where Eric? Can I leave my answer like this? No What now do I need to put on top because there's nothing there? Now I have to put the one in but I waited until I had to I was kind of hoping the letter would end up there. You can see ryan. What do we do? We're avoiding negative exponents. I get rid of them here and I never bring them back Don't need to see Is there an exponent outside the brackets? Yes. Oh, it's a fraction So that exponent's going to go on to everything on the top and everything on the bottom It's going to go here here here and here I'm going to get this Three to the negative two x to the negative six All over four to the negative two y to the positive four now what Elevator Does this stay or does this change levels? And do you mind can I write it as a nine right away because I'm pretty sure that's what it's going to become Does this stay or does it change levels? Does this stay or does it change levels and I'm going to write it as a 16 on the top if that's okay Why do the fourth stays where it is? Is there any simplifying I can do? Does 16 over 9 reduce? No. Are there any x's on top or y's on top? Oh, this is done One line of work D I see two big brackets. Is there an exponent outside any one of them? Then let's rewrite the one with no bracket. I'm just going to say okay Let's rewrite this bad boy here. This is a negative an eight an x to the eight a y to the fifth A 25 an x squared a y And this is going to be This two is going to go on to everything so it's going to be a 25 or the 25 come from Five squared it's going to be an x to the sixth a y to the 18th all over What's a negative squared Gone because that's really a negative one in front of the four. I treated it separately So a negative squared is going to be gone a four squared 16 x squared y To the tenth and we're going to go right to the final answer in one fell swoop Really? Yeah. Hey look closely. How many 25s on top grand total? One how many 25s on the bottom grand total How many left and where Right, I just saved myself a bunch of math because I noticed that Can you do that mr. Do it? This is all being multiplied now So this is one big fraction on top and one big fraction on the bottom. Heck yeah Let's do numbers So the total numbers that I have on the top is what oh first of all how many negatives on top grand total One how many negatives on the bottom grand total? Is my answer going to be negative or positive? Oh, don't use red mr. Do it back to blue Let's do the numbers the top numbers simplify to an eight See it the bottom numbers simplify to uh 16 See it What is eight over 16 and lowest terms? By the way, do you think if there'd been like a three here like other numbers? Could you have handled figuring out what the top number was just by multiplying it in your head and what the bottom number was? Yeah Now you're ready How many x's on top grand total see him 14 see him How many x's on the bottom grand total? Four how many left and where how many y's on top grand total 23 Yes How many y's on the bottom 11? How many left and where? Think about that. We did that in one line. It's not that I think my method works pretty good I don't know why they don't teach it that way. Yes Isn't what y 12 Isn't what y 12? I don't know. Let me see how many y's to the uh 23 11. I think 12 Ellen you get a candy later I can't do math. I'm so bragging about doing all the math in my head. Apparently it was the grade one math that tripped me out subtracting Turn the page. I think turn the page or did I miss a question? No Okay The last skill by the way, I don't know what happened when I sent this printed this it looks like It chopped off the edges, but it didn't chop off the questions. I'll fix this another time Today's notes. It doesn't look like the the page number seemed to have got chopped off, but you know where we are Says changing the base What was the base of the exponent? I think the base of the exponent was the number that was in the base of the down here One of the skills we're going to pick up this unit if you want to be able to take one base and write it as a different base And that's also why I had you write all those exponents on the previous page all those powers It's easier to show you than to try and explain how we're going to do it. It says this Convert each of the following to the base indicated write that as base three. What this is saying is this Instead of a nine to the two x I would like a three here. How can I write a nine as a three? Is that still nine to the two x? Except now I have power to a power See it If I simplify it Not three to the four x. I have just rewritten the base nine as a base three Believe it or not That's actually going to be useful. We're going to be able to solve certain equations that we were unable to solve before b 125 to the two minus x rewrite that as base five okay The exponent is two minus x but instead of 125 right there I want a five to a five to what? I heard three and I heard four which one It's five to the third now what oh Power to a power and I guess it's going to be three times that and three times that I think my final answer is going to be five to the six minus three x Apparently five to the six minus three x is the same as 125 to the two minus x c Eight times 16 to the x power to base two by the way You can't go eight times 16 and get cell eight times 16 is uh 200 no you can't 128 sorry, but you still can't Why because that would be violating bed mass that's got an exponent on it This one doesn't get to the exponents first instead I'm going to write this as a two And the two to the x How can I write an eight as a two? Too cute How can I write a 16 as a two? I'll give you a hint two To the fourth Now I can take this and simplify a little bit further. I have power to a power yes So I can write this as two to the third Times two to the four x I can do one more thing though. What's the base in the first bracket? What's the base in the second bracket? Are you saying my bases are the same and I'm multiplying? What do I do with exponents when my bases are the same and I'm multiplying? Ah This is actually two to the three plus four x Apparently that's the same as eight times 16 to the x. This is kind of fun. Mr. Do it again. I It's not really mr. Okay D this To base two Okay, it's gonna be a three x Now I didn't tell you to memorize this one everybody put your pencils down Everybody get your fingers out Close your fingers ready with me two some four eight 16 32 double it Double it 120 double it 256 double it 512 is two to what power This is why I didn't have you remember the twos are the most common ones and technically you should know the first nine You know what if you can just double things and most of you can you can get there two to the ninth Really that's going to be one over Two to the 27x But we're not done. This is not base two Technically you could argue and I would agree with you Caitlyn. This is base a half Because the two is on the bottom It would be wonderful if there was some type of operation that caused it to change levels like an elevator as it were How could I get the two to the top? Introduce a negative exponent This is the same as Two to the negative 27x because I would have elevated it to get it down there. There that's a base two Not a base one half the last one Probably the toughest one, but nearly cool. What do they want me to write this as base? What? Two over three They want me to write it as the base of a fraction Which probably means they gave me a fraction in the brackets. Did they? Can I write the 16 as a two? Can I write the 81 as a three? That's not so bad In fact I'm going to argue this first of all. I would probably write the x plus five because I feel better. It's no longer blank 16 Is two to the what? 81 is two to the what? This is or is three to the what? Four did you say the same exponent for both of them? See, I think I can write this two-thirds to the fourth is that not 16 Over 81 and they did that on purpose They wanted to make sure I had the same exponent for the top and bottom because otherwise How could I possibly write it as a two-thirds if I had different exponents this wouldn't simplify properly? Joel do I have a power to a power now outside the brackets? Do I have an exponent and inside the brackets? Do I have an exponent? Say yes So I'm going to go like this No, no, and I'm going to get two-thirds all to the power of 4x plus 20 by the way, you know what the most common mistake here instead of writing 20 would a kid's right? They put a five because they're in such a rush. Yes, I'll have that as an answer to pick from on the multiple choice In fact, I'll put that as a the first answer you spot Check your math So there is a review of exponent loss with mr. Dux handy dandy shortcut for never having to deal with stupid negative exponents Because I don't think you really need to And a quick review of not review quick brand new of changing the base of objects. What's your homework? um I'm gonna go one a one b one d one e ab the Two is good Okay, we're with me now like I seem like I've lost some of you already stay with me I'm skipping three. Whoo-hoo 4a is good 4b is good 4c is good 4e you I like that 4f I like that I skipped d I really need to slow down really come on stay with me here. I'm calling them out Number five. We're on number five. So you should have turned the page by or be on the next number five I think I like Well, let's see. We got easier ones and nastier ones. You know what? I'll go five a c e and f six all eight Oh, wow, and I gotta gotta do 10 Fractions and negatives and brackets. Oh my yeah, look with it It'll be fine. It's not as bad as you think