all right 4.3 multiplying and dividing rational expressions okay so good news is this one's actually not gonna be very different from just simplifying like within the last one so for example let's do let's do well till way back let's say if we had an expression like this if I add like three over four times and let's say 2 over 2 over 6 right we were multiplying these guys well well multiplying is the easy one right it's not like adding a subtracting I subtracting you have to figure out that old common denominator and everything but with this one it's just it's just straight across right so this isn't too bad this would just be 3 times 2 is 6 you know 4 times 6 is 24 and then you could simplify that really okay well you know I can simplify the top side by the bottom there so divided by 6 divided by 6 end up being 1 over 4 right but we can do this same thing I want to show you a little easier way here too that we could have done that we're gonna do a lot more now you can just multiply that straight across if you like but the other one that probably a lot more valuable here is I can see that when you're multiplying you just have stuff in the numerator and you just have stuff in the denominator and when you're multiplying it's like they're one one thing this that I'm looking at right here would be exactly like 3 times 2 over 4 times 6 it's the exact same thing right so there's just stuff in the top and stuff in the bottom right so right from step one we could have started canceling things cuz I can say well if I look at like the 3 and the 6 here then I could say well you know I could have just cancelled 3 like 3 over 6 would be the same thing as 1 over 2 or 8 like that would become a 2 on the bottom and then I can see I've got a 2 on the top and a 4 on the bottom so I could cancel those that's like you know one that began with one that had become a two so then in the end all I have left is I've one right just that one over and then two times two is four it does the same thing but this way we're gonna use a lot more especially because we're dealing with variables now so let's throw some variables in because I think I make more sense I'll stop a raising them now I'll keep it on here so yeah so for for these ones what we're gonna do is it's gonna be something like this where it says simplify alright so we're going to simplify these so number one so it could be a question like this B over 12 times 3a over 2b squared okay so now like I said I can I can multiply straight across and the top and bottom if you like but that's gonna make things a little like to be fine we can then you cancel it after but I'm just gonna start canceling things now all right I've got a B on the top over here and I've got a B on the bottom over here I can start canceling that right that's the same thing as like B over B squared even though they're you know on other sides of the equation there's just a B on the top over here and B and the square in the bottom over here so all of that would become well the B would cancel with like one of the B's there so I've got like one be left on the bottom and then I could take this three right there's that three there in that twelve I can think of that as being like three over twelve so I could I could simplify this I can cancel that three and so this would be three over twelve would be the same thing as one over four right that's all become become before and then just write what you have left so this will all be the same thing and typically this is the way you want to do it you want to work your way down for these expressions is so that so then in the end we'd say that that we had we would have just a or a have just got that a left on the top over and then four in a two on the bottom that would become eight because of multiple and then I've still got to be and that's it so it's a lot like simplifying right or just just got that multiplying in between there so I let's do no number two so let's say we had six M cube times n plus 1 over 15 M times n plus 1 there's all of that so normally right from the last lesson we could just cancel those but you're our whole expression we're gonna make it big along here so it's X there's our not X that's gonna be x and then 2 N minus 1 over m times n all right all right so this would all be the exact same thing as with like without the multiplication symbol right it would just be all this stuff on the top like everything on the top on both sides and everything on the bottom on both sides right they could all be like sandwiched together if you like we could even write it that way maybe I'll do that maybe for this one right we could cancel right now they'll show you that it would be the exact same thing would be the exact same things if we wrote it like this I can just multiply the 2 and the 6 right now so there'd be 12 M cubed times n plus 1 and then n minus 1 that's just like all the stuff on both sides all sandwiched together over and then the bottom we'd have 15 M oh sorry 15 M and there was M times M I guess so we could say that was M Squared and then we still had min and we had an N plus 1 okay now this is straight up from what we did last lesson right so we start canceling so I've got like an N plus 1 up top and then then plus 1 in the bottoms they cancel I've got and M cubed up top so that would cancel two M's in the bottom and you'd be left with one M on the top and then I've got 15 over 12 well 3 goes into both of those so I can divide the top by 3 is gonna be 4 over 5 and then I've still got an M up top and I've got an N minus 1 and then on the bottom I've still got that in so there you go that'd be a hopefully that's making sense for you guys listen let's do another actually you know what and while we're doing this too we should be talking to learn on permissible values so remember not permissible values they're saying what is M and n not allowed to be so the only time we run into a problem here it's with our denominator so what is am not allowed to be let's do em first M is not allowed to equal and I'm asking myself what would M have to be in order to make the whole thing equal to 0 well it can't be 0 right because if M itself if I threw a 0 into that that'd be 15 times 0 which would be 0 and then that 0 times this is still 0 right that would make everything 0 so M is not allowed to be 0 okay and then the other part of this would be n so n is not allowed to be well if I look over here right it's not allowed to be 0 again because of that so it can't be 0 but then it also can't be I'm looking at this one this time right if n right there if that was negative 1 well then that'd be negative 1 plus 1 that would make this 0 everything in brackets there which would make my whole bottom zero so an is not allowed to equal negative 1 so those would be my non-permissible values there okay hopefully you guys find it more room than me let's do let's do another so three okay so let's say we had x squared over six instead of a time similar I'm gonna go like a dot right so I don't get confused with the X's I was already put in there so 2y over 3x okay so again I could sandwich them all together but really I can cancel these just the way they are so let's do the X's first so I'm gonna say okay well I've got x squared and I've got an X down there so I can cancel like one of these X's right so well the squared would cancel with the one on the bottom and and then if I look at my six and my two there right that would simplify to 1 over 3 so everything up top here I've really just got X Y that could say 1 if you like over and then you know so that was one and that was a three right so then 3 times a 3 that's already down there is just 9 all right now so that what we were doing here so far was when there was already kind of set up for you or anybody that the factor anything so now what we're gonna throw into it is factoring as well so we're going to factor these things before we get going so if we had this question let's say so we had two x squared minus 12x over 15 x times 5x over X minus 6 okay so this is one of those weird situations where I can't just say okay well there's an X and there's an X I'll cancel those right you can't do that and the reason for it is because this is a binomial right there's like two different pieces I've talked right it's both of those over 15 X so I can't just pretend it was only in that one all right so have to what we have to do here is we have to factor everything to see what's actually common to the top so I'm gonna factor the top there so we'll say well what's common to 2x squared + 12 X okay well two x's so we're gonna write that as 2x I'm gonna start my bracket and then write what I had left so after I factored out 2x out of two x squared right I've still got X left and then - and then 12 divided by the - I took oh that's six so that's my top and then the bottom I can't factor that so I'm just gonna write it again so 15 x times 5x over and then this worked out nice X - 6 and that that is like it's in brackets right that's that's under that's all over that right so that worked out nice because now this is free game I'm allowed to cancel this whole X minus 6 and X minus 6 okay so they were in the same way I can also here just do the one of them there I can also cancel this X with well one of the two X is up there I can cancel this one in that one and what else so now we've got the 15 over or search 5 over 15 so I could cancel that 5 and that would become 3 on the bottom and then if I just write everything I have left here after I've cancelled it all I've got left 2 x over and then I've just have 3 left on the bone okay and then technically I should say mine on permissible values mine on permissible values here would be well X is not allowed to equal and I want to look up here right now I look there and I want to look there before I cancelled things so because of this one X is not allowed to equal 0 right because that was 0 then I'd be dividing by 0 so it can't be 0 and on this one X right there is not allowed to equal 6 because if it was six that'd be 6 minus 6 they'd be 0 so it can't be sick okay so there's my answer for that and alright now let's do let's do some maybe a little bit tougher ones where we got a factor a little bit more so one okay so let's do serums looking very good example here there we go alright so now what was that that was four will do five here all right so let's say we had X plus 4 over X minus one times X plus one over and then here's what's gonna change up we've got x squared plus 5x plus 4 all right so these ones I would leave right you can even put brackets around these if you like cuz it's kind of nice to think about it as like a it's like a thing right you just got like that's like one packet that's like one thing there it looks like I could almost cancel like the X minus 1 and the X plus 1 but they're not the same so I can't cancel them but now the bottom right that doesn't look quite right but what I can do and what I have to do is I've got to factor this first to see if there's anything I can cancel out of this so we're gonna write it all again we're gonna write it as X plus 4 over X minus 1 times X plus 1 and then I'm gonna factor the bottom here so the way we factored these ones right where you've got a squared and then X and then number in the end is they just both get a bracket right bracket bracket and they both get next and we're looking for numbers that multiply to 4 and add to 5 okay well that would be 4 and 1 alright didn't and then they'd both be plus and then that worked out well because now I can see well X plus four would cancel with X plus 4 X plus one would cancel with X plus one and all of that cleaned up really nicely because now in the top now this throws kids off too right I've got like nothing left up there so this is where people would be tempted to say my answer is just X minus one but that's not right right because X minus one is in the denominator right that's in the bottom so I've got nothing left up top but there really is something there there's really a one right there's still a one even though we cancelled everything out there right so there's one and then it's over X minus one so that would be my final answer there and then again for the the non-permissible values here we should we should've been stayed in those two you do that the beginning as well but again what I want to do is I want to look at it before I cancelled things and I think I want to look at it right here right because it's actually a lot easier to see what my non-permissible values would be so I'm going to say okay X is not allowed to equal and then I can see from this one it can't be equal to one I can see from this one it's not allowed to be equal to negative four and I can see from this one that's not allowed to be equal to negative one so those would be money non-permissive values it can't be equal to you could write it like this like 1 comma negative 4 comma negative 1 it doesn't really matter but you could also write it like this where it's 1 plus or like plus or minus 1 & 4 and negative 4 so either one okay so you guys will have to do that factoring in these questions it's not it's not too bad usually usually you'll know because you know you'll factor it and then things start cancelling you'll know you probably did it right when things you're able to cancel one or two things all right so that that is multiplying now I haven't done division yet but division is actually gonna be there in more room nope I do that okay so vision is actually gonna be pretty easy as well so let's do like a little review of dividing fractions in general so let's say if I had three over four divided by let's go let's go three over two right those dividing those so here's the here's the thing to remember dividing by a fraction is the same as multiplying by the reciprocal I know that sounds confusing but what that means is that if I'm dividing by a fraction that would be the exact same thing as going 3 over 4 and then changing it to a multiply so so x and then the reciprocal are these two numbers flipped upside down so that'd be the same thing as 2 over 3 okay so then at that point you could say okay look at 3 over 3 those cancel the 2 and the 4 but that would all just turn into 1/2 right so that would just be with a standard fraction you guys did that in math 9 now we're gonna be doing rational expressions with it so that's taking that same idea now let's do one like this we'll say and then Ruby on 6 so if we had 12 x squared over 15 and now it's divided by 3x over 2y okay so now we're gonna say okay we need to well this one's not too bad we just change it right off the bat so we're gonna change this to well you keep the first one the same don't flip the first one because it's just you change the second one so we're gonna keep this as 12 x squared over 15 now I'm gonna make this a multiply but then these two guys flip so this will be 2y over 3x and then it's just like the last things right now it's just like the question we just did so okay I've got x squared up top so the squared would cancel with the the X and the bottom and I've got well really just this 12 over 15 3 goes into both of those so we still are left with so 12 divided by 3 is 4 I've still era I'll just do 2 steps for there's still the 2y up talk over and then you've got 5 divided by the 3 that we did there would be 5 and I've still got the 3 on the bottom and then the X is gone right oh sorry I forgot that the X up top they were still an X on there it looked like a crossed though they were still an X up top okay so that would all be equal to 8 XY over 15 that should be it okay and then technically though we still again we should have been doing our non permissible values right so let's go back up and see what's not allowed to equal what so if I went right to the beginning right let's go right back up to the beginning so I'll say well there's nothing even in here right so I'm gonna worry about anything like dividing by 0 there but right here I can say okay well why not allowed to equal 0 right because I'd be that was 0 I'd be dividing by 0 but here's what the other tricky part about this this is the 1 remember with division is after we flipped it right so it wasn't allowed to be this because we couldn't divide by 0 but watch what happened after we flipped it now the X is on the bottom right so X X is also not allowed to equal 0 because now it's in the bottom there right so when you're doing division you have to take into account on the second term look on this term you have to take into account like what is X or Y not allowed to be on the bottom and also on the top there because the top on this one is gonna become the bottom on the second on the second move there around the second like after we flip it okay so you got a look at both there okay all right last one we're gonna do one more question here and then and then we're done ok so that was six seven alright so we've got x squared minus X minus 20 over X plus 4 and then divided by x squared minus 3x minus 10 over X minus 1 okay so now now we could flip this right now and write it all again but you know I'm gonna do I'm gonna kind of flip it am factor at the same time just so we have to write it all like twice or like another two times but so they're a mess start factoring as I go here so I'm gonna factor this top one so that would be someone's gonna write my brackets they both get an X and now I'm looking for numbers that multiply to 20 and add to well one right there really is a one sitting there okay so multiply the 20 you know add to one I'm just gonna so well four and five right so four and five which is promising because the difference between four and five is one which I want and I need to multiply to a negative so one of these guys has to be negative and one's got to be positive all right so it's got to be this one right this has to be negative 5 and positive 4 because they need to add to negative one okay and then that's over X plus 4 and then I'm gonna change this right now I'm gonna change this to multiply and then flip these guys right I'm gonna factor at the same time so on the top we've now got X minus 1 and on the bottom I'm gonna have this this part there alright so that'll be well I'll put my brackets down I'm looking for numbers that they both get next I'm looking for multiple numbers that multiply to 10 and add to 3 so probably five and two and then one's gonna be negative one's gonna be positive to get to that you know multiply to negative 10 so this will have to be negative 5 and positive 2 okay so we're looking good here there's lots of stuff to cancel so now well actually yeah well we'll come back and do the non-permissible values after so we'll cancel now so now at this point I can say well I've got X plus 4 on the top and X plus 4 on the bottom so those cancel nicely and I've got X minus 5 and X minus 5 on the bottom so all of that cancels and in the end all we have is X minus 1 over X plus 2 that's it and then you'd say where X is not allowed to equal so where X is not allowed to equal and then let's look back up here what was not allowed to equal so from the beginning right there right X is not allowed to equal negative 4 right it can't be equal to negative 4 there so can't be equal to negative 4 looking at this X is not allowed to equal 1 because of that one but then here's where a change right because normally I wouldn't even care what was on the top at the beginning but that got flipped so now it's on the bottom here so now I do have to take this into account it's not allowed to equal negative 2 because of that one and it's not allowed to equal 5 because of that one so it's not allowed to equal positive 5 so there you go there be your non-permissible values for that one okay