[Music] [Applause] right so in this video we're gonna have a look at rationalizing the denominator again make sure you've got some paper making some notes on some of these before you ever go so rationalizing the denominator just means that we want to make this denominator here okay a whole number and not desert anymore it's all it means and so we are going to create an equivalent fraction so the same fraction but where there's not assert on the bottom thinking about this in another context if we want to make an equivalent fraction a fraction something like 1/2 we can times the top and bottom by 2 or anything we like and that would make two quarters okay now these are equivalent fractions but they're written in slightly different ways the fraction on the left has a two on the bottom the fraction has a right has a foot on the right as a four on the bottom so we're going to do the same with this we're going to write an equivalent fraction but it's not going to have a root 5 on the bottom anymore okay so there is one thing that we can times the bottom by that root 5 by to ensure that we gets a whole number so if I rewrite this ok there we go so there's some thing we can times the bottom by to make it a whole number and we've seen this before we can times it by root 5 and that would make it read 25 which turns into the whole number 5 so that would work and we're going to apply the same logic to all of these questions sometimes the top and bottom by root 5 to make sure I see an equivalent fraction so on the top we get 3 root 5 and on the bottom we get root 25 which is just 5 okay and again you can write that working out down there for the bottom if you want okay 5 root 5 times root 5 3 25 which keeps us 5 so there we go that is our final answer there we've rationalized the denominator it's no longer a third on the bottom it's the whole number 5 let's have a look at under the 1 ok so a little bit more length in the wording here rationalize the denominator of 3 over root 6 giving your answer in the form at root a over B where a and B are integers so let's have a look what happens when we do the same process here then so we've got a root 6 on the bottom so let's times the top and bottom by root 6 to get rid of that cert on the bottom so times root 6 times root six there we go so we get three root six on the top and on the bottom we get root 36 which is 6 okay so this is where this extra part the questions going to come in then because it says to write it in the form roots a over B now at the moment we've got this 3 in front of the root 6 so that means this fraction must simplify so if we think to the side I mean let's forget the root 6 exists for the moment let's think about the fraction 3 over 6 similar to what we've got at the moment but without that root 6 on the top now this fraction simplifies the top and bottom both divided by 3 it simplifies down to 1/2 so we can apply that same logic to this one I'm gonna divide the top by 3 I'm gonna divide the bottom by 3 I'm gonna simplify it so we get 1 root 6 on the top don't have to write that one we can just write root 6 and we get 2 on the bottom it's a root 6 over 2 there we go that's in the form that the questions asked for a is 6 B is 2 and we've managed to simplify that as well so do have a look out for this if you can simplify okay so look another okay last question here before you ever go rationalize the denominator of 1 over 8 root 8 and given your answer in the form root 2 over a what a is an integer ok let's write this out 1 over 8 root 8 let's apply the same logic then solicit times the top and bottom by root 8 see what we get so on the top one times root 8 is root 8 and on the bottom we have 8 root 8 times 8 so that's 1 root 8 there isn't it even though we don't write that so 1 times 8 is 8 and root 8 times root a is root 64 or 8 so let's work this out slowly so 8 times root 64 is 8 times 8 so I'm gonna get 64 on the bottom there so let's have a look what we've got food just rewrite that and we get root 8 over 64 okay it's still a not in the form that it's asking for though because the question does say write it in the form root 2 over a so this question does simplify and that's because root 8 simplifies so root 8 can also be written as root 4 times root 2 which years tea routine so actually that simplifies down I can write that as 2 root 2 over 64 again still doesn't quite match the question there I've got this scenario like in the last question where I can divide the top and bottom by 2 because we have two root two on the top 64 on the bottom so if we divide those both by 2 we get root 2 on the top and 32 on the bottom and that's our final answer there that was quite a long one for this type of question is quite a lot going on there we had first rationalizing it here then we simplified the bottom and in this separate step of simplified the top we could have done that in one step really here simplified the bottom simplify at the top and then we got our final answer which we could still simplify so that was quite an extensive one there but let's have a look you'll can either get a few questions just make sure you're following this process here looking at rationalizing that denominator but x in a top and bar by whatever that root is okay so here's some things I would go out make sure you pause it and have a go and here they are suppose there and I'll go over the minutes a ok so rationalizing this first one times the top and bottom by root seven and we get to root 7 over 7 final answer and so next one times the top and bottom by reaped in and you end up with 5 root 10 over 10 okay now there's no hint in these questions here but you might have spotted the top and bottom they're both divided by 5 so if we divide the top by 5 we get root 10 and the bottom by 5 we get 2 okay but as a side note that is actually rationalized the question didn't say to write it in a certain form but it does actually simplify down which is always worth looking out for and this next one rationalized 4 over root 8 so let's times the top them by root 8 and this root eight keeps coming up here root 8 isn't actually fully simplified so we could simplify that to start with but I'm just gonna leave it see what happens as times top and bottom by routine so we get 4 root 8 over 8 okay well straight away we could simplify that we could divide top and bottom by four and we get root eight over two but then again we should really spot that this root eight here on the top isn't simplified although that is rationalized it was rationalized at this point here where we had the eight on the bottom but always looking to simplify with fractions so we could have left it is four root eight over eight but more than likely a question is going to want us to simplify it down so we have root 8 over 2 and then root 8 is and I'm gonna do this to the side square root of eight is root 4 root 2 which is 2 root 2 so root 8 on the top is 2 root 2 over 2 well this just goes even further now the top and bottom both divide by 2 so actually all of that ends up is just being root 2 there's a lot of steps there so although we did actually rationalize it here you can see if we simplify it all the way down it goes all the way down to root 2 times in the top and bottom by root a gave us that then we could simplify this fraction which that enabled us to get retake over to then we can simplify this root 8 into 2 root 2 and the top and bottom then divided by 2 just to get us all the way down to the bottom there I quite an interest in 1 right and then the last one here times the top and bottom by root 6 let's rewrite this 3 over 6 root 6 soap it's not as many steps as the last one times the top and bottom by root 6 let's see what happens so on the top we get 3 root 6 root 6 doesn't simplify thankfully for this one and then on the bottom we get 6 lots of root 36 6 times root 36 which is 6 times 6 so we get 36 on the bottom now again not as many steps here but that one does actually simplify the top and bottom both divided by 3 so till dividing the top by 3 gives you root 6 and dividing the bottom by 3 gives you 12 there we go and now is our final answer there right okay perfect that's a look at the next bit okay so similar process to the to the ones before but we've got two pieces now on the top so I'm gonna don't we use the same logic we have times at top and bottom by root 5 and just see what we get ok so now we know on the bottom we get 5 root 5 times root 5 is root 25 which is 5 I'm on the top now we've got 2 times both these pieces so really we've got something like our single brackets that we looked at so it's something like 1 plus root 5 in the bracket multiplied by this root 5 so the root 5 times 1 gives us root 5 copy the symbol plus and then root 5 times root 5 gives us 5 maybe case we rationalized that doesn't simplify top and bottom hope they don't both divided by 5 just be careful this whole top part here doesn't divide by 5 even though this 5 here does root 5 does not divide by 5 okay so it doesn't simplify is our final answer there and let's have a look at another one okay here we go so 3 minus root 2 on the top 2 root 2 on the bottom let's do the same thing again some sort of motton by root 2 and see what we get it's on the top root 2 times the 3 3 root 2 and root 2 times this negative root 2 gives me negative root 4 or minus 2 perfect and then on the bottom 2 root 2 times root 2 is going to give us a notice to the side it's gonna give us 2 lots of them because this is 1 root 2 so it's 2 times 1 which is 2 root 2 times root 2 root 4 to big 4 which is 2 times 2 which just gives us four lovely so on the bottom four and there's our final answer three root 2 minus two and four right okay here are two questions to each other go up so pause the video have a go and go over the answers in a second ok the answers for these and so the first one times two top and bottom by root three on the top we get to root 3 minus 3 over 3 and there's your first one so the second one times the top and bottom by root 5 we get 3 root 5 plus 5 over 2 times root 25 on the bottom here we're gonna get 2 root 25 just 2 times 5 so 10 on the bottom and there are your two answers okay we've got one more question to look at well this question doesn't look very nice okay so what I've done is I'm gonna go over one this one on the left and I've got one for you to have a go at so over here we've got 12 over and on the bottom we've got another fraction plus root 5 so 1 over root 5 plus root 5 on the bottom and some one look if we apply the same logic here if we terms the top and bottom by root 5 let's see what we get tops easy enough we get 12 root 5 so look what happens on the bottom I've got two pieces here 2 times by root 5 so looking at this fraction to start with that times a fraction by a number or by a by a root in this case it's going to multiply that denominator so actually what we're gonna get for the fraction part when we times up by root 5 is root 5 over root 5 plus I'm gonna times this root 5 by root 5 which is 5 so plus 5 ok he doesn't look very nice now just I'm gonna look at this fraction here root 5 divided by root 5 well anything divided by itself just equals 1 so every 5 divided by root 5 is gonna be 1 so actually what we've got on the bottom here in total is 1 plus 5 and 1 plus 5 6 it's not as have a look we get 12 root 5 over six and again this actually simplifies so the top and bottom there look both divided by six twelve lots of root five and six on the bottom so dividing the top and bottom by six we get to root 5 over 1 okay we don't need to write there anything over one day we can just leave it there as 2 root 5 and there's a final answer right it's one question on the right there to have a go out so pause the video have a go and then I will go over in a sec okay so the final question here times in the top and bottom by root 3 on the top we get 4 root 3 and on the bottom times in that fraction we get root 3 over 3 sorry root 3 over root 3 add 3 so let's simplify that down we get 4 root 3 over let's see what's on the bottom there on the bottom we have 1 plus 3 which is 4 and the top and bottom now both divided by 4 so top divide the top by 4 you get 1 root 3 divide the bottom by 4 you get 1 and again we don't need to write that sore answer for that is just root 3 all right there we go that's the end of the video we're going to look at some slightly harder rationalizing on the last surds video so again if you like this video and it's helpful please like please come and please subscribe and I'll see you for the next one