A LEVEL MATHS: Binomial Expansion - Positive Integer Powers LESSON 3

hey guys just to let you know that you can catch these lessons live every Thursday from 6PM until 7 PM UK or BST time and I'll just allow you to ask any questions that you want and get some answers to them without further Ado let's get on with the lesson okay welcome to the third I think uh a-level lesson that we're going to be doing uh today we're going to be covering binomial expansion so binomial expansion is uh somewhat a tricky topic because you can do the whole thing on a calculator so on that part it's kind of easy the downside is that it's very easy to make a mistake as well so let's go through what Banner will expansion is so when we expand brackets right so let's say we have X plus one the power of 2. okay then all we do is we do X plus one X plus one multiply them out we get x times x is x squared and in the middle we'll get a plus 2X and at the end we get a plus one right cool now what about if we had X plus one cubed okay well okay this gets a bit longer so X plus one and then I won't skip ahead let's actually do it the whole way you would expand two of the brackets so this one which would give you x squared plus 2X plus 1 and then you just get X Plus 1 again okay then you're going to expand it one more time so X cubed plus x squared 2x squared plus 2X plus X that's one then you'd gather the like terms you have X cubed plus 3x squared plus three x one so as you can see it's taking a you know it's taking a bit more effort right um so what we need to do is find an easier way to do this now can you imagine if you had to do X plus one to the four okay what about X plus one to the 10. so it's getting a bit tricky right but you might notice that you have these patterns coming up so number in front of x squared here is one then the number in front of X is two it's one okay then you have one three three one and when you did X plus one to the four you can get numbers as well which I'll show you in a second now we can actually work out what these numbers are by using something called pascals triangle and it's really easy to write this out you start with the one and then below you put two more ones then you're going to go out one more and you're going to write one and then one and what you do is you add these two numbers to work out what this middle number is so one three one then once again you draw ones on the outside one plus three gives me four three plus one gives me four so you have one four four one okay and you can keep following this to basically create uh what the numbers are going to be in front of this so this would be for when the power is zero obviously this is going to be one then when the power is one we just get one and one right because what you'd get is you'd get X plus one so I have one is just X plus one so the numbers in front of these are one and one then n equals two we get one three one and so on now as you can imagine if we're doing x to the power of 10 we have to go 11 rows deep that's pretty tricky right like if we kept going it'd be longer and longer longer and would get some ridiculous ridiculously long row and obviously if you're in an exam and they ask you to expand X Plus 1 to the power of 10. you don't do that okay so instead there is something else that we can do okay and there is a button on your calculator that says NCR now if you have the Casio calculator which is recommended that you use which is the FX 991ex which is this calculator here can you see the button it's actually above the Divide sign which is NCR to get it you have to press shift and then the NCR button now this button does something very very interesting so I want you to keep an eye on this line over here okay so the N part of the NCR is the power of the bracket so in this case if we look at the cubed the power is always three okay and the r is what place you are so are you looking at the first term the second term the third term the fourth term but and if you're doing computer science you might recognize this counting starts from zero one isn't the first one so just to reiterate n is the power of the bracket and R is the position you're in and it starts from zero okay so when I type this in it's going to give me the number that is in front of the first term which is one that's correct then what I do is I change this to one I'm looking for the second term it's three Now look for the third term it's three and the fourth term is one one three three one which is exactly what we get here so now all I need to do is change the power of X can you see that okay so when if I go back to zero I can write that as one times x cubed then go to the next one along plus three and then all I do is I reduce the power of x and make that two again three and then reduce the power of X and then plus one okay so if I were to do X Plus 1 to the power of 10 this is what I would do I would get my calculator now n is the first number we change that to a 10 because I'm looking at the power of 10 and the first digit is represented by zero press equals and I get one okay so I'd write one and then X is always to the power we start with the power of the bracket to the 10. then what I do is I change this to a one and I get 10. so what I do is I do plus 10 and I reduce the power of X by 1. and then I go along again and I do two you get 45. and if you wanted to look up the answer you'd find that this is indeed correct and I reduce the power again and then plus dot dot now the trick is here is that in the exam they won't ask you to write out the whole thing okay they'll ask you for the first three terms or four terms or you just have to do a few terms they won't ever ask you to do 10 11 terms okay if you were to look up with the expansion of Xbox one to ten is you would get these three terms at the beginning so that's pretty damn useful right instead of having to expand more and more brackets or instead of having to use Pascal's triangle I can just use the NCR button now there is a small caveat okay it's not just this simple what if this wasn't a one but instead was a three this is where things get more interesting okay let's say we're doing X plus 3 to the power of 10. well again I'd use the NCR button so these numbers will stay the same it's fine the power of X will also stay the same but now you're probably getting a bit suspicious how can everything be the same well technically speaking this number on the end here also goes in a bracket but it starts off as power of zero which does nothing and you increase this power while this power goes down so I said x to the line here now it's going to be 3 to the power of one so that should actually be a three but in this case it was a one so it'd be ten times three then 45 uh x to the power of 8. so I've reduced the power again so I increase the power again so instead my answer is now going to be again it's still x to the 10. but now it's going to be 30 x to the nine and 3 times 45 would give me 115 not bad so 45 times 3 would be 135 sorry x to the A okay so if this number isn't the same all you need to do is make sure you're increasing the power of the number while decreasing the power of x and now you might also ask well what if it was a 2X not 2x just by itself obviously plus uh three it's five ten well then you're still going to have the numbers in front but then instead of x to the 10 it becomes 3x to the 10. so you need to do 3 to the power of 10. so I hope that that is okay and you're okay with that so that would be how you do that one okay so now when we go to the questions let's just do a bunch of examples because this is the best way to learn in my opinion they sometimes word it in ascending powers so first three terms in ascending you're going to have your constant so like a one then you're going to have X and then x squared that's what ascending Powers means okay so instead of doing it this way well we've done the highest power we've gone down we need to go backwards now going backwards is not that tricky okay so what we're going to do here is we're going to use the NCR button once again okay so what is the NCR button when we have the power of six so here changes to a power of six and the first digit oops is going to be a one so I'm going to put a 1 . now with ascending powers the constant goes here and it has the highest power okay so technically that means X over 2 is to the power of zero okay so it's just literally the inverse of what we did up here where we went down with X up with the number we're going down with the number up at X it's fine because it's sending Powers okay then plus and let's bring our calculator back what the second term be so again even though it's a one it's the second term because we start counting at zero six so we're going to write six and remember we do reduce the power of the two so from six we're going to five and increase the power of this so X over 2 to the power of one okay good now bring back my calculator now we're going for the third term which is represented by two fifteen so fifteen two again reduce the power of the of the two to make it four increase the power of the X to make squared now let's expand this what is anything to the power of zero it's one so we have one times two to the power of six so guess what I'm going to do I'm just going to use a calculator I don't you know don't you know you don't have to do the whole thing 2 to the power of six enter the exact same thing and you'll never get it wrong equals 64. wonderful so that whole thing equals 64. Plus now look I'm not going to rewrite the whole thing use the arrow keys to edit what your thing was so I change that to a six I change the power to a five and now what's the actual value inside here well it's one over two so I'm going to leave the one over two there 96 and then remember that there's going to be an X and again I'm not going to change anything I'm going to press the right arrow go here it's 15 now instead of five it's four and instead of the power of one I'm going to put the power of 2. 60 that is your number in front of the x squared that's it that is four marks four marks I did the whole thing on a calculator how much brain power did I use all I have to do is remember reduce this power increase this power reduce this power increase this power that's it is mind-numbing how easy that is okay it's really not that tricky at all now the tricky bit is you know doing Part B but part A which is four marks two-thirds of the marks of the question is that simple okay let's just go down it then says using your expansion find an estimate for the value of 2.05 times 10 to the power 6. so if you notice this power here is going to be the same as this power here always okay so what they're saying is we have the expansion for this right and I want the expansion for this so what I want to do is I want to know what do I sub in for x in order to get the estimate and what we do here is we work out what x is so the first thing I do is I take the that was a horrific I'm taking the sixth root of both sides which gives me two plus X over two equals 2.05 subtract 2 from both sides I get X over 2 equals 0.05 times both sides by 2 I get X is equal to 0.1 so all I need to do now to find out an estimate for what um two point zero five to the power of 6 is is I sub in x equals 0.1 now keep in mind there is an A2 version slash year 13 version of a binomial expansion so this only works so this method here is for positive integers okay it says for POS so where N is a positive integer so a natural number so what you'll find is next year or later on we're going to deal with n when it's a negative integer or a fraction or fractional negative integer and you still have to do the kind of um find an estimate for that is a key part of this um topic so just for those of you that might be curious as to where these binomial expansions come from we use them to find estimates because sometimes there are math expressions that are too complicated for us to work out using computers and obviously we can't do it by hand so instead we use binomial expansions to approximate and estimate the answer okay because if we keep going in this expansion X becomes smaller and smaller and smaller right because at first X is 0.1 what about 0.1 squared that's 0.01 then cubed 0.0001 to the power 4 0.0001 as you can see it's getting smaller and smaller and smaller which means we're actually getting closer and closer to an answer which is actually pretty damn good okay so if you any of you do physics or maths in at University you'll be using binomial expansions a lot then all we do is using our calculator we Sub in exactly what it says 64. plus uh 96 plus 60 and I get 74.2 okay now what we can do is actually see how good this estimate is and this is actually again something that'll come up later is you'll have to work out as a percentage how good your estimation is that's pretty good right we're off by two hundredths so if we went again we would get closer and so on okay so that's one example if that's all okay so obviously the next thing we can do is just keep going through more questions and uh yeah this gets a bit more interesting okay but the first question is still four marks and it's still relatively easy okay so what we're going to do once again is use our fancy MCR button okay so for a once again we're going to do shift n on the left hand side is our power which is seven right hand side is what position we're in so the moment we're in the first position which is zero so we're going to do one two to the power of 7 and I'm going to keep writing the minus like the other bits these bits here I'm going to keep writing it even though you don't need to okay don't need to and then that'd be plus then what is the value going to be seven oh that's handy reduce the power of the two and increase the power of the X and it says first three terms so we go one more term 21. reduce the power of the two I think we spell this cool remember the next step was just subbing in exactly what it says right so let's do that in fact I want to be really pedantic I'm going to put exactly what it says in one two five seven you don't need to put the X minus one over eight to the power of zero 128. I'm using absolutely no brain power right now okay I'm not going to put a plus or minus because I can see there's minuses now so it might change change that to A1 is that to A6 back to a 7. see bear in mind four marks means it should take you four minutes to do this but how long is it actually taking me to do it bear in mind I'm actually doing this I've got a bit of a handicap because I'm I'm clicking to type this in if I was doing this by hand I can use both my hands at the same time instead of just using one hand like a peasant 21 over ticket I'm done that's it four marks probably the easiest four marks you'll get but you need to make sure you're doing it right enter it exactly on your calculator you cannot make a mistake it's impossible I dare you to make a mistake okay so let's have a look-see they tell me that we have f of x because seven cool and what they're asking us to do here is given that the first two terms are 384 not minus 104 X find the values of A and B so what you need to do here is you need to expand this bracket so if you have a look we've already got this kind of expansion right now what we can do here is replace the second bracket with what we've just worked out okay that's all I've done is just replaced it so make sure you're okay with that okay it then tells me the first two terms so we're going to have x to the zero or constant plus x to the one that's the only terms I care about okay so when I expand this I don't need to fully expand it if you get ax cubed that's pointless we only care about the numbers I.E the constants and anything with an X so I'm going to do ax times 128 . now do I times x by anything else no because if I times it by X I'd get ax squared I don't need x squared if I times about x squared I get X cubed I don't need X cubed so I only need the first terms okay cool now B times 128 okay that's fine and then minus 56. BX do I times it by the last thing no because then I get x squared I only care about these two bits so again you wouldn't be wrong if you fully expanded it but there are two issues one it's a waste of time this is half as many terms as I should get I should get six two you'll get confused I now know that everything I have here is useful if you expand it half your stuff is not useful and you'll see the thing oh my God what do I do so it's telling me that the first two terms 384 minus 104. X so from here you might see what you what we need to do okay what we're going to be doing is equating x to the zero terms and equating x to the one terms separately so let's start if we start x to the zero x to the zero in this is just 384. make that a colon actually and what is it in this term it's just 128 B so in other words b equals 384 level one two okay so again like don't risk it like even if it is basic math just do 384 two use a calculator now equate the X1 terms we have minus 104 you don't need to put the X's but I will put the X's just so you guys are clear 128 ax minus 56 BX now if you notice I can divide this through by X that's why I don't need to write it I can also Sub in for this B is 3. so in other words I have minus 104 equals 128 a again let's use our calculator what is minus 50 whoops what's minus 56 times 3. minus 168. so plus 168 to both sides we get um 64 or again you can just use the calculator I'm being lazy even with a calculator in front of me which is kind of funny 64. equals 128. hey divide both sides by 128 again you probably know what the answer is already but I'm going to use a calculator just to make a point okay aha and that there is also four marks okay which I don't think is too bad once you know what you're doing the issue is knowing what you're doing okay any issues with that okay let's go to the next one then this next one is a bit of a pain in the back side because we have to fully expand it which is kind of annoying but I'll show you it's actually not that bad okay so once again we're going to use our trusty calculator NCR button the power is five with some from the first place so this is actually interesting these aren't numbers so technically I don't need let's do the pile thing so like it'd be one p to the five Q to the zero plus and then so on so you don't need to do it but I'll do it just to kind of make a point great typically make a mistake immediately P to the four again just reduce one power increase the other power Plus 10 P cubed Q squared I have a lot of trouble writing cues I don't know why again another 10. see and if you notice they're always kind of palindromes right they're the same forwards as backwards which is and I won't say it's interesting but something that's it finally it's a bit tedious but not hard because I got a bit bored during it and I'm sure you did too so if you want we I'll write this out like properly so to speak so oh all right one day I won't be stupid there we are and then Plus there we are so it then says using the last two terms of the binomial expansion or otherwise so we're basically given a probability of date being late is 0.1 and let P represent quality they've actually done a lot of work for us believe it or not using the last two terms of the expansion find the probability that Dave is late no more than one time in a week so p is late okay so if I have a look at this right if you remember your probability right the probability of him in being late once is 0.1 probability of him being late and then late again would be 0.1 times 0.1 or in this case it would be p and then P times P so P Squared yeah and so on so forth okay so for him to be late no no more than one time he's either not late not late right or he well in this case he'd be late or late late right now if I have a look at this I'm looking for the terms where there are no P's because he's not late at all so that's this term or there's only one p which is this term that's it that's why they gave you this hint using the last two terms of the binomial expansion okay because in this one here he's late twice early three times late and so on I hope that makes sense because when we come to statistics uh it's going to be um we have binomial distributions and things like that where this can somewhat come in so it's kind of useful to know so if p is 0.1 the probability he's late is 0.1 then the probability is not late has to be 0.9 because probabilities add to one okay so now we're just using the last two terms so be five p is 0.1 Q is 0.9 and again I'm going to use a calculator so again just because so some kind of actually 0.91854 cool so that's seven marks we did that very quickly even though I was explaining it to you um and it really wasn't that bad it's just they try and trip you up using the second part so as you may be aware at least 10 of your marks have to be something called unseen marks which basically means they have to give you a situation you haven't seen before but it's the same maths this is one of those parts generally speaking the last part of any question is going to be this unseen stuff so you may never well the people who did this exam first would never have seen a question like this they would have had to do it themselves like think on the spot okay so your goal by the end of a levels is to be able to think on the spot okay and be able to kind of pick apart these questions quite easily by yourself okay so practicing them is really really important um but yeah so 10 has to be unseen the reason for that is if 10 wasn't unseen you could just learn mark schemes to get basically 100 in your exam so they put these 10 marks in to say that you know it kind of separates the people that are really good at maths and understand it versus the people that are just really good at studying maths so yeah um they also put this P plus Q to five in it doesn't change anything the first things there and the second thing's there and you change it normally this is a number and an X but what are p and Q well they're just numbers we don't know yet so what they work exactly the same even if they use like Theta and Delta still letters doesn't matter anyone have any questions about that good to hear excuse the next question then so up to an including x to the power of three would be the first four terms okay so once again so play and we're going to do our NCR button shift into our the power is eight and the first position is zero please please please remember it's zero basically in your binomial expansion your first coefficient is always one okay and we're going into ascending order so we want one to the power of eight and then again you don't need to write this part but I'm going to keep writing it just to make sure you are aware it's there and it's easier to not forget either eight again you might be noticing patterns in what the numbers are as well that's good you can use that to help verify and check you're doing it right still use a calculator but check that it's right check it makes sense so it doesn't make sense I want to re-look at it again reducing the power of the one increasing the power of the of the 4X and the power should add up to this power here right and we're up to x squared so we need to do one more okay so again it's tedious boring but to be fair if you find this tedious then you'll really enjoy the A2 binomial expansion where you have to do it kind of without a calculator so just uh keep that in mind again I'm going to type in exactly what we're doing oh there's one I'm absolutely shocked to this one again you can do this a lot quicker on your calculator by hand and always remember the X's as well don't forget that because that's not on your calculator although in some calculators you can actually put x's in and everything which hopefully you can see how powerful this tool actually is well for those of you that are just doing math to get into your degree option then you probably don't care but 3584. there we are done and once again show that you can again approximation okay cool um so again with the approximations this is actually gonna be really accurate because it goes up to X cubed generally speaking in maths and physics we only go up to x squared sometimes X cubed we don't even go further than that so we're actually getting really accurate answers but let's have a look again this expansion is for this thing here so we have one plus four x the eight and we want to work out what we need for X to be in order to get an approximation so again we're taking the eighth root this time so that'd be 1 plus 4X equals 1.04 subtract one 4X equals 0.04 divided by four x equals 0.01 so if you want to do that over here our approximation for zero four approximately equal to 1 plus 32 times 0.1 plus 448 0.1 squared plus 3584. again let's just type it into our calculator so this whole topic really it's use your calculator but then you also need to use your brain to apply it to situations so generally speaking students find these to be free marks okay because they find it really easy and they say that once they've done a bunch of practice questions at home when they're nice and comfy but guess what happens every single time they go into an exam they make mistakes okay so they're not subbing the right thing onto their calculator they're not do some other random thing and I'd speaking of mistakes I've made a mistake I subbed in 0.1 instead of 0.01 see so even though it's boring and tedious you can quite easily make mistakes like I just did okay and I'm wondering how many of you watching noticed I made a mistake and how quickly you noticed because that's kind of exactly how your brain will be right that gives you one point and it says to five significant figures so we're gonna go up to that second three so one point three six eight four okay easy to make a mistake okay again I'm wondering how many of you actually managed to spot that I made a mistake you know because I intentionally did it of course but um very easy to make a mistake and as a result students go in saying Dynamic extension is easy I don't need to practice it and yeah it's easy but when you're in an exam you're under a hell of a lot of pressure right so your end of year marks or your end of your actual exams if you're in year 12 or 13. it's very easy to make a mistake so writing out exactly as I have it's longer sure but it can really guarantee you won't make mistakes so just be very conscious of that and how easy it is to mess something up okay okay so enough lecturing let's go on to the next question so again as you can see there are actually um different styles of questions so this is the other startup question you can get which is where they'll basically ask you to um they'll put like an unknown variable other than x and they'll give you what the coefficient of one term is and then say okay find the value okay very very very common if you look through any past papers this is the most common so most common Style question okay so like we did before where we had a and b I believe it was that's also extremely common as well so it's like there's an unknown variable it's always it's the most likely one to come up okay so get good first four terms which means X cubed right of the expansion two plus KX five six okay let's do this once again what is the power the power is six what position am I in at the moment I'm in position zero and it's a one I'm absolutely shocked and it's ascending power okay which means we're starting with two to the power of six KX to the power of zero into one six again it's the first four terms that's another thing by the way normally they say the first three terms okay so I've seen students in questions where it says list the first four terms only do three terms which means you lose an entire mark so just like be very conscious of it so that is a five by the way two and then reduce the power of four so what I like to do is when I read the first four terms if you notice I immediately wrote X cubed right so that in mind also we're lucky it gave us the coefficient of x cubed so if you only did three terms and you saw it said X cubed you'd probably be reminded that you need to change question a but normally they give like x squared or the X term in which case you're kind of screwed if they um if you made a mistake in the first part um wait did I just do this one yeah I did so zero one two okay so now you might bond in what do I enter in my calculator because I have K's well I never bothered writing the case and the uh the X's so if it's a KL so don't bother writing those what I'm going to do instead is just literally write actually makes my life a bit easier to enter this in so that's going to be 64. Plus change that to a five so 32 and then KX don't forget four let me get 16. and now it's going to be K squared x squared bear in mind and then lastly cubed K cubed x cubed so again remembering the powers is probably something that may trip you are um and so that's just working out the powers of two so if we were to work out the actual values here so again I'm just working out the paths 2 initially um so to work out the four powers you do that so this would be like 192 for example so 64. okay so it can help to kind of break it down into doing the separate bits separate but it's completely up to you what you want to choose 160. okay then fobby so again this is four marks pretty dang good in my opinion uh not much work but it does require a bit of concentration which again can be lapsing especially in an exam probably the coefficient of the X cubed term is minus 20. so what we have here is if we look at the X cubed terms we have 160 K cubed x cubed equals 120 but it says coefficient so we get rid of X cubed coefficient is the number in front of x so fairly easy K cubed is just equal to 120 over 160. so K is therefore equal to so it's actually negative 20 isn't it oops uh the cube root of negative 20 over 160. and all you do is uh calculate that value okay so again we can negative a half that's our answer not too shabby at all okay so I can just uh have a think about that make sure it's okay okay okay so let's do another one so yeah this is another style of question they can ask you again it's fairly similar to what we've done before but let's have a look see I just saw you aware um I'm actually going to start uploading shorter summaries of lessons because I think these long lessons can be a bit tedious for some of you so there'd be five minutes ideally where it would literally summarize the chapter so give you like the individual equations and then there'll be a link saying if you want to watch more then you can go here to find um exam question practice okay but that is the plan for the future but let's start with this so again find the first three terms and send a power bank Okay cool so NCR button again so the power is five and the position is zero it's one shockingly so we have one one to the power of five minus two x about zero plus five one to the power four now because we've reduced the power minus two x to the power of one and it's the first three terms so we're going up to x squared okay that's ten one spot three minus two x to the power two cool now we need to do is actually Sub in this stuff into our calculator so we have uh one one to the power five that makes one absolutely shocked at that uh that's actually not gonna be a plus this time is it so what we do is go from left to right change that to a five reduce that power increase this power -10 X 10 cubed 40. squared okay and then it says so find the first year here of this thing here first three times again now we should know that that means it's up it's going to include x to the 0 x to the one x squared so we're going to do first is sub in our expansion that we just did which is 1 minus 10x minus 40 x squared and then we're going to expand these brackets but only going up to a maximum of x squared that's the key part that um students sometimes get wrong so this one is going to times everything in that bracket so we have 1 times 1. minus 10x minus plus even 40x squared then the x is only going to times the first two things because the third thing I'll make it X cubed so we have plus x minus 10x squared now simplify 1 minus 10x plus X gives you minus 9x plus 40x squared minus 10x squared gives me plus 30X squared I'm done bonus three marks in not too long I think okay so yeah again again with these lessons um it's good for you to join in live because that way you get to ask the questions that you want so you get to suggest topics and everything like that but it can also be good to like see me work through them live obviously the lessons are kind of edited and they're updated um upload it again afterwards so you can kind of watch it in your own time but with binomial expansion especially it's a lot of practice okay so and also having kind of a correct format so it creates a layout is really important as well so if any of you have any questions then please let me know why you're doing the live stream but I will be uploading shorter summaries which then link back to this as well but is anyone in the chat that is stuck or doesn't understand this if not I will do another question before we finish okay let's do another one then so again uh Don't Be Afraid don't be kind of weirded out by the fact that the marks look a bit weird this is actually from the old spec but keep in mind the old spec still has all of the same stuff in it if you look this question looks exactly like our other questions right so let's do it when the first four terms that means we're going up to x cubed of this expansion so what we do is we use the NCR button to work out what our coefficients are so n is always the power of the bracket which is seven and R is where we are in the expression so the first place is always zero so it gives me one so I write one then what I do is I write since it's ascending I'm going to look at the 2 first so 2 to the power of 7 and then KX to the power of 0. okay then I'm in the second place so I put this as a one and I get seven so now I get plus seven and what I do is I reduce the power of the two so instead of seven I make it six and I increase the power of the X and that's all you do there's rinse and repeat Until the End okay so once again change this to a two because we want the third position 21. reduce the power of the two to five increase the power of the X what do you do for the next one well I'll use the NCR button to find out what we're doing that's 35 and then I reduce the power of the two increase the power of the x is that simple you could go on and expand the whole bracket if you wanted to that's as easy as it is okay but now we need to actually work out these values on this is where some students mess up so I write the exact thing it says here so 1 and normally I would actually include the bracket of the X in it but because we have a k which is an unknown I can't I'm not going to do that so first thing is 128. Plus and what is I change the number in front seven four four eight now it's KX again I probably did that the wrong way around but that's totally fine again if there was a number in front of X I would also add that bracket on at the end like I done in the previous questions but in this case doesn't really make much sense to you and again you need to be really good at this because there is A2 variant of this which is a bit more painful okay I'm just looking at this and that my friends is probably the easiest full marks you can get okay in an a level paper tells me that the coefficient coefficient means the number in front so all of this equals 1890. so in other words 560 K cubed equals 1890. now divide both sides by 560 always divide by the thing in front first you get K cubed equals 1890. I have a 560. then you cube root both sides and you get k is the oh awful looking cube root 1890 or 560. guess what I'm putting into my calculator the exact thing shift key root fraction three over two that seven marks you can do it in about three four minutes and honestly it's not tricky it's easy to make a mistake so no questions then I will end it there because we're practically on time to finish but again we've done quite a few questions today and again if you want to go back and and kind of pause and like look through the answers and do them yourself and everything that might be a good thing to do but if there are no other questions then I wish you all a very good evening and I'll see you next week