 So questions from the homework anything you'd like me to go over. Yes Cara Love to do numbers. Is that the one of the P's in the Q's the big nasty one that I gave yep I'll save that for last anything before number 10. Yeah 5f Yeah Okay, 5f6c and a and b Okay, that by the way I am gonna say five and six are really the skill that you need out of this lesson the exponents review was important because We're gonna be looking at how exponents behave in a very different way for the remainder of the unit And it'll make sense if you fit it into your already learned exponent review about rules But what we're really gonna be doing quite a bit of this changing basis. So 5f I Looked at this and I said well they want me to write this as base four over three The problem is the four is on top and Carly. That's a three cube on top by the way Hopefully you're starting to memorize some of those exponents and That's a four on the bottom Four cubed, but that's a three on the bottom over here How the heck could I get the four on top and the three on the bottom? Well, first thing I did was this I said, okay, this is three over four cube To the X plus two Is that okay so far kiddo? It would be wonderful if there was something that I could do that would act like oh an elevator as it were That would somehow cause a number to say change levels Where as an exponent? I Think if I instead of putting a three right here as an exponent If I put a negative three right there that would be automatically causing The four to move to the top the three to move to the bottom and now I have it as base four over three Do you follow that little reasoning there? I did it in one step here I should have made it out one. Oh, and now I can go. Oh, I have a power to a power. This is all gonna end up being four-thirds to the power of negative three X minus six So if you ever try to get a base to match what they want and it's Reciprocal the top it is on the bottom and the bottom is on the top Elevator introduce a negative inside your exponent and that will cause the numbers to change levels Normally we got rid of the negative to cause them and cause the number to change levels But in this case I have to introduce one Is that okay? 5c and I'll let you try a and B on your own 6c sorry, it's a 5c 6c Ready Eric? What base do they want me to go here? What's one quarter as base four four to what power is one over four? Thank you. Yeah, this is definitely four to the negative one What 16 ignore the one over for that 16 is four to what power? One over 16 is four to what power? Yes That's the first thing I would do Then I would say hey, I have a power to a power. Mr. Dewick says take care of that first almost always so this is gonna end up being four to the negative one times Four to the negative eight plus two X. Is that correct Eric? I think yes. Yes. Yes I Looked in the bow. You know what there's one more thing I can do. This is what my answer doesn't quite match the back What's my base right here? For what's my base right here? You're saying my bases are the same Am I multiplying then what do I do when my bases are the same and I'm multiplying? What do I do with exponents back in grade nine? I can rewrite this as Four to the and when I say Adam what we're really doing is we're gathering like terms There's no X's over here to add to the X's over here But a negative one take away eight is a negative nine plus two X although I bet in the back They probably wrote that as two X minus nine because that looks Okay, that's what you're trying with these ones as well So if I'm writing that that's gonna be a three squared and a three cubed and I'll get rid of power to a power And then I can actually add the exponents because they're the same base 10 C no 10 This is far harder than anything you'll come across which means it's good practice Again the goal is we try and find the test easier. I hope Hmm oh Power to a power mr. Dukes said Always do that first. I got an exponent outside the brackets. So this one half is gonna go On to there and on to there and on to there and this one fifth is gonna go on to there and on to there Okay It's gonna be 64 to the negative one half P to the what's a positive times a negative Carol? It's gonna be a negative up there. What's two times a half? Or what's half of two? What's half of two? Don't shrug your shoulder girl. What's half of two? Thank you. That's what two times a half is it's half of What's a negative times a negative Kara So it's gonna be cute a positive number the real question is because your brains are dead right now and you've forgotten your fractions How do I multiply two-thirds times one-half? I took care of the negatives Multiplying fractions is the easiest of all the fraction operations Multiplying fractions is very simply top times top bottom times bottom That's it top times top two over bottom times bottom two over what? Ready top times top two over bottom times bottom. What's three times two girl? This is two over six in lowest terms. What's two over six? This is gonna be to the one third Multiplying fractions is the easiest fraction operation both top times top bottom times bottom Adding and subtracting is when you got to find the stupid common denominator thing Over Kara have you got a graphing calculator yet? Okay, otherwise I was gonna show you another way that you can do this on your graphing calculator. Anyways um Letter P down here Kara Negative times positive it's gonna be negative five times the fifth you ready It's really five over one because five is a fraction Times the fifth, you know multiplying fractions is the easiest operation. It's top times top bottom times bottom You get five over five which in lowest terms is with the negative still kicking around q to the Positive times negative is negative. Let's see if I can do ten over one times one over five without writing it out What would ten over one times one over five be in? Lowest terms, please. I know you can I think to yes So step one is getting to that line right there Then mr. Doeck said the next thing we always do is elevator The 64 to the one half is gonna move to the bottom This P is gonna move to the bottom the q to the one third stays there This P is gonna move to the top and this q squared is gonna move to the top is that okay? pizza cake now Are you ready? By the way, I noticed how many letter P's do I have on top one? How many of the bottom one? How many left? How many P's on top one how many on the bottom one how many left? none Let's do that. We feel better now. I Tried to hint this to you last day I said one half as an exponent is one worth remembering to the one half power means something very very specific it means square root Memorize one half and one third what you want to do from now on is if you see one half as an exponent But I also want you to right away see square root of that number or if you see one third as an exponent I want you to also to root of that number because this is really the square root of 64 which in your head the square root of 64 is ah That's gonna be an eight down there now, you know where that eight came from in the answer By the way look up wrong. Yes Wrong. Yes Did the piece cancel? Wrong. Yes. I already I know the answers D. I like I've been trying to tell you in multiple choice questions often You don't need to do the whole question. However Are these two bases the same? So I can add the exponents. What is two plus a third now here? I do need a common denominator Yuck instead of a two I'm gonna write is that still a two six over three is that technically still a two and Do I have a common denominator? You know what when I add the exponents six thirds and one third is you know How many thirds six thirds plus one thirds how many thirds? Did you say seven hard for me to read your lips. Yeah, that's why so to be what? Q yeah, that sorry After all that about getting the piece to cancel. Thank you. I need to mind my peas and queues Is that okay Am I gonna give you one that I know What I expect all of you to follow that explanation. Yeah, like I'm hoping most you're going Oh, I could have got that. I just didn't think right stubborn and clever is the phrase you've already heard me say a few times Let's jump to Where I think math 12 truly begins This is probably brand new unless mr. Gerard mr. Kim was he stuck it into math 11, which is possible? I don't know lesson two Which is page 99 and 97 Woman number one says reviews simplify that by converting each term to a common base Did I do a question for Eric where we converted and for Carly where we converted some stuff to a common base? Can we call that our review? What we're going to look at today are two new types of equations The first is an equation that has a now. It says a rational exponent a rational exponent another word for rational is fractional exponent What we're going to look at today a star is equations that have fractions as exponents on the X's Like example one Can you all at a glance recognize what I mean? That has a fractional or rational exponent? That has a fractional or rational exponent. They kind of stand out Vlad stay here. I'm going to scroll ahead real quick Hey, that's got a fractional or rational exponent. So does that so does that they stand out This does not that's not a fraction. That's going to be a different kind of equation But when we have a fractional exponent What's the procedure? Well, here's what it says The first thing you do is you raise both sides to the reciprocal power of the exponent. So what? Watch example one says solve for X in the following X equals Negative four-thirds for X to the negative four-thirds equals 81 bear with me someone at the door Someone's gonna bring a big carton to your Cheerios or something. Yeah, there we win. Okay ready Here's what it said to do raise both sides to the reciprocal power which sounds confusing, but it's a really easy trick This is a great trick What's my exponent right here? Isabel read it out to me. What's the exponent? What's the reciprocal of negative four over three negative still what's the reciprocal of negative four over three negative? I'm gonna put This side to negative three over four hang on I'm gonna try writing that so you can see it a bit better this side to negative three over four and This side to negative three over four gotta be both sides because remember your graded equation solving Ellen you have to do the same thing to both sides all the time plus five both sides by both sides by two both sides So the reciprocal power now. Why is that so nice? watch Ready Kara my love You were the one who asked me this early this will work out perfectly. I'm kind of glad you asked number 10 leads into this Do I have a power to a power over here? Say yes So I multiply the exponents. What's a negative times a negative? Positive and then how do you multiply fractions top times? Bottom times can you see you get 12 over 12 see it? Which is what in lowest terms? In fact, don't write this down. It works out to that Do I really need to put a one there or isn't there automatically always a one automatically there? In fact by doing that the X is by itself. I drop the equal sign down Now on the right side, I have 81 to the negative Three-quarters, okay Now what what would I do with that answer? Well, not yet. I never said to do flower power first. I told you there was always something I did first Elevator first I would go like this One over 81 to the positive three quarters Now I would yes, I agree flower power as it were strangely enough This is gonna be one over The what root of what to the what? It's out of the cube root of 81 to the fourth or the fourth root of 81 to the third fourth root of 81 to the third and I'm just gonna move this over so I have more room to write perfect No calculators because I would feel comfortable giving this to you on the non-calc section of your test and I've already told you I'll say it again. No, your test is gonna have two parts a non-calc and a calc Calculator section Cover up the three in your mind. What is the fourth root of 81? Three cubed I heard it There you go, except why don't I continue in black so that it just makes more sense Why change colors probably through a question when do I do that when there is a fractional exponent both sides to the reciprocal power? Be is there an exponent? Yeah, now first of all, what if they're just been a squared there? What would you do this one you already knew how to solve there was a squared there? How do you get rid of a squared? Square root there was a cube there. How do you get rid of a cube cube root? So that one we showed you in grade 9 and 10 and 11 now we're saying what if there's a fractional exponent there? No problem put both sides To the what's the reciprocal of 3 over 2 and if you do that These will always cancel because it's always gonna be 6 over 6 or 8 over 8 or 12 over 12 It's always gonna end up being an exponent of 1 In fact in this one you're gonna get 3x minus 5 all to the one don't need to need to write all to the one equals 27 to the two-thirds Now what? No negative exponents. So now what the stupid rhyme that I gave you flower, but yeah, right it as a I mean technically Right it was a radical. Okay, so I'm gonna read drop the 3x minus 5 down And this is the well, I know the 27 goes there the what root of what to the what? cube root of 27 squared if I'm evaluating these in my head I never do the exponent I cover up the two in my mind. So right now I don't see the two cube root of 27 squared This is 3x minus 5 equals 9 and now I'm gonna argue that's math 8 Terms of level of difficulty. I think in math 8 is when you do the plus in it How would I get the x5 itself here plus 5 to both sides? Divide by 3 x equals 14 over 3 The only problem Joel is this is a little awkward to check in your head. You could Go back and check your answer. Certainly. I probably could for This one here may be But the message is be a little uber careful a little more careful. Don't make sloppy mistakes So the first type of equation is where the exponent is a fraction The second type of equation is called an Exponential equation an exponential equation is where the x is an exponent We have never done those before ever We have never given you x's up here till today Never given you x's up here till today. So here it is a brand new equation How does it say to solve this? What's our approach here? It says use the following procedure first thing Right both sides with the same base. That's why Eric. I snuck that in last lesson second thing we're going to do is we're going to equate the exponents and Then the third thing we're going to do is solve. It's much easier to do it than to explain it example to a Justin, what's the base on the left-hand side? What's the base on the right-hand side? Do I have one base equals one base say yes are the bases the same? Then I can reasonably assume. Here's our reasoning Ryan If five to something equals five to something the somethings must be equal to each other If five to something equals five to the something else, even if they look different They must be the same we call that equating the exponents if I have one base equals one base And my bases are the same I can equate the exponents and I'm going to argue Ryan. That's math eight That that equation that we end up with How to get the x by itself Can you see the answer is going to be two in your heads or if you can great in your homework If you can solve these in your head and get the x by itself good We'll show our work in our notes so that when we're studying, you know what the heck you did Minus three actually. I'm not showing my work. I'm just showing the step aren't I and then divide by two Joel what's our strategy try and write it as one base equals one base and if the bases are the same we equate the exponents Are you ready? Let's look at B. Joel my friend Are my bases the same right now? What's my base on the left What's my base on the right up? It's not three hundred and forty three at least that's not what I see. I See seven to something I can rewrite this as seven to the x minus two equals three us three Seven to the third power this this this is why I told you you had to memorize those exponents that I gave you last day On that previous page that we listed out. This is why you need to know these powers Maria do I have one base equals one base say yes Are my bases the same say yes Then I can equate the exponents on my next line. I can say hey x minus two has to be three Maria what does x equal in your head showing no work? With authority, please without hesitation and a loud strong voice negative five No, no, no, no, don't I plus two to both sides? five right Yeah, did you say negative five or did I miss your I'm sorry, maybe Okay, five right right don't make that don't do the math 12 right and then trip on the math eight We don't want to do that There's so much fun. Let's do another one. Okay Mitsu, what's my base on the left hand side? What's my base on the right hand side? Don't say 81 because it's not I don't see an 81 there What is it? Ah This is really three to the five x minus one equals three to the fourth To the three x I need to tidy up the right hand side a bit Trevor The four and the three x I need to tidy those up This is going to be three to the five x minus one equals three to the mitzvah. What does this simplify to when I Not just 12 12 x Mitsu my friend do I have one base equals one base? Or my base is the same then I can equate the exponents My actual equation that I'm solving is five x minus one equals 12 x Okay, now we're grade nine because there's x's on both sides How would I solve this I'll tell you what I would not do I would not minus 12 x from both sides that would be silly Because I might be leaving with a zero on the other I'd get confused I would minus five x from both sides, right? And I'll get negative one equals seven x Mitsu get the x by itself how ah, thank you negative one seven Really you need the calculator Ah, no calculator put it down I would give you this on the non-calc section of your test no problem And since you walked right into my snare Brett, let's pick on you my friends if you need to be awake anyways What's my base on the left hand side? Now on the right. I see a 27 and a root 3. I'm going to write those all as threes This one already is so I'll just drop it down What's 27? I agree What's the square root of 3 as an exponent? I think mr. Dewick went ballistic about 10 minutes ago telling you to remember this That's when you see root 3 you always instantly also want to see three to the one half. They are interchangeable Do I have one base equals one base say no I don't I got two bases still on the right hand side two threes. I better combine them What do I do when I'm multiplying and my bases are the same? What do I do with the exponents? So this is going to be 3x to the hmm What is three plus one half as an improper fraction Now you lost about half my class. Thank you very much for not letting me down For the rest of you who suck at fractions brett is at least mediocre at fractions But the rest of you for those of you who suck at fractions look up You have to find a common denominator to add fractions and three over one is six over two And so brett said that's the same as three to the seven over two it is Now do I have one base equals one base? Or my base is the same Then I can equate the exponents and this one's kind of nice at the end because I don't have to get the x by itself It already is Is that okay bre? Do you want to get a little lower ellen you can I got a table? It's a little shorter. I can dig up if you want to Or if you want to lay down totally good with that Okay Turn the page D is much more like the level you're going to get on your test. I like D. I like E I like D. I like E. I'll be completely honest a b and c too easy I don't even call those c minus so Take a close look at D What base do I want to write both of these as? three three So let's see now. I'm going to be way more careful. I'm going to show more work. This is a much more difficult question I'm going to write this as Three to the third to the x minus two equals The exponent over here is going to be x plus three What's one over 81? Three to what power? Ah elevator good There's the 27 and there's the one over 81 Is that okay eric so far? Now power to a power This is going to be Three to the three x minus six And this is going to be three to the Negative four x minus 12 Alex, do I have one base equals one base or my base is the same? Then I can equate the exponents I think carly I would plus four x to both sides And at the same time I would plus the six over to the other side because I can do both of those steps at once because I'm at 12 Plus the six over Carly how would I get the x by itself? There you go one more We are for me. We are done. We are complete Trevor glanced at the clock and said wow that was fast. Hey time flies when you're having fun doing math How can it not it's like being in disneyland. It's just never enough time What base by the way e this I would consider b plus maybe even a minus So the first one I would consider c plus ish Reasonable level this one I would consider a little trickier What's my base on the right hand side on the right hand side? What's my base? Question Eric What's my base on the right hand on the right six over five? I'm pretty sure that's what I want to put on the left uh-oh On the top that 125 I can't write that as a six Uh, I can't write it as a five On the bottom that 216 currently I can't write that as a five Wait a minute. This is an awful lot like the question that you asked me from the homework Where we said well it almost works except we want the top to go to the bottom and the bottom to go at the top Back we want the numbers to go on an elevator as it were perhaps ah So How can I write this as six over five? What is 125 to the 216? Well, first of all five is three sorry 125 is five to the what? Three 216 is six to the what? Three, so what do I want to put here? Not quite a three Ah a negative three And then the x the negative x over four drops down and this is six over five Is that okay, Ellen? By the way, this is a negative three what's negative three as a fraction negative three over what what's any number always over One because I'm going to be multiplying two fractions together Kara. You know how I multiply two fractions together It's the easiest operation top times bottom times In fact, I think we're going to get this Six over five to the power of Three x over four Is that right negative times negative positive top times top bottom times bottom equals six over five To the three x minus three Spencer. Do I have one base equals one base? Are my bases the same? Then I can equate the exponents I can write this as three x over four equals three x minus three I'm just gonna look into your voices. Stop fraction. No, relax Relax relax Hopefully in math nine you learned that when you have a fractional equation There's an easy way to make all the fractions vanish Remember how? What I'm gonna multiply everything by the denominator. I'm gonna multiply everything. I'm gonna multiply everything I'm gonna multiply everything by four by four by four and I always when I'm multiplying Put it in front because if I put the four on this side What will it look an awful lot like an exponent and now it's gonna confuse me because I'm in an exponent unit right now So when I'm multiplying I always put it in the front Why is this so nice Maria because here I get how many fours on top one How many fours on the bottom one how many left and where none? In fact, I get three x equals 12 x minus 12 I'm minus 12 x from both sides And I get negative nine x equals negative 12 I get x equals negative 12 over negative nine, although what's the negative divided by negative rand And 12 over nine the lowest terms Now you could conceivably check that answer This is why I said though we've reached the level we're going level We're going back and checking answers is almost more trouble than it's worth The message I have for you is just be really paranoid and careful when you're doing these building sloppy mistakes Check your work So two new equations today Justin fractional equations also called rational equation And exponential equations Quick pencils down What we're going to ask ourselves now for the remainder of the unit is okay I get this exponential equations mr. Dewick where the x is up as an exponent. That's pretty good so far I'm stalling right now by ah there boy. I hit that quite a while ago and that program wasn't opening up What we're going to ask ourselves is what if instead of this? What if instead Can I write a five as a 12 or a 12 as a five using exponents? Not yet And to get around this we're going to have to invent an entirely well Define an entirely new mathematical operation called the logarithm the log button on your calculator that you may have seen and wondered What the heck it was for? Having said that hugely useful to be able to solve these equations we are going to do some very cool applications We'll look at half-life. We'll look at population growth. We'll look at richer scale and earthquakes All of these are what are called logarithmic scales or exponential growth questions and you can do some really cool Some really neat stuff But that's in our future. However, it's going to take me about five lessons or four lessons of Careful definitions and walking you to a point to get to this point homework Well, number one is rewriting stuff as a common base. We did that last class two is good three is good four except look up Cross out F. Well, you know what I take that back I'll let you try it F though Not on test. I'm giving you that because in about Christmas time Something similar to this will be what we'll be doing in trig So I'll let you think about this one and wrestle with it and anybody who can figure it out on their own We'll get a candy next class if they're the first ones to tell me about it By the way g and h G still fair game h may be a little overkill in terms of Level of difficulty on the test this I would consider Okay, but nasty this I would consider a little bit past nasty Okay Six is good Seven and eight There it is now if any of you have calculator and or workbook funds, I would love to