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yes I used to work at my downloads and this is my old uniform from a decade ago now I'm a math teacher and today I'll show you guys to find out if 2019 to a fourth power plus what early 2019 primed or not first of all let me tell you guys it's actually pretty hard to show if a number is prime it's actually much easier to show if a number is not a prime because we can just try to factor this if we can factor this to get a product of two numbers and you also to make sure these two numbers are not 1 then we can come to that this right here is not a prime then we'll be done especially we notice that we have this to the fourth power Haho have the fourth power it's actually factorable for some of that and let me show you how so here we go we will make some observations go to some algebra and some exponent or that let me begin with this will keep it because we like it 2019 to the fourth power we like this for this power dot 2019 power no not so much I also want to have a four spot right here that's how can do that first of all this right here is an odd number why don't we look at this s1 plus 2018 and we test that we can look at this as plus 4 to the first power times 4 to the 2018 and yes I squeezed our coefficient but in fact this is your ok and the truth is this poor actually help us to factor things out more nicely as well you'll see but this is not the fourth power at all and you notice that yes 2018 even now but it's not divisible by 4 it's ok because we can look at this 4 as 2 to the 2nd power and then this right here is raised to the 2018 power so you multiply this 2 with 2018 and the new power is of course this times that which is 4036 and now this is a test of what for right and as you can see we can write this down again as 2019 to the 4th power and this is plus what well I want to do kdss something to the 4th power this right here all i have to do is divide it by four so the inside power will be 1009 and don't forget we still had the base two right here so two to this power to this power is two to this power namely this yeah so here's the deal in fact this right here it's factorable and I'll show you guys how to do it well aren't you sure it is the general case because the general case is actually pretty famous and this one here actually has has a name this is called East or V German identity and of course this is the name of the mathematician and she says when you have the expression in the following form namely a let me just write down a to the fourth power plus four as a coefficient in the front times some other thing that's a B to the fourth power like this this right here is actually factorable and you might be wondering how is this factorable because back in the algebra days your teacher told you the sum of two squares it's not factorable with real numbers well this ratio technically you have the fourth power if you're willing to this thing about how can you make things happen then you actually make you know good things happen let me show you how first of all this right here I don't know how to do it I know like wouldn't it be nice if I have an expression that's easier to do it and we know when we have that expression in this form something squared plus two well things like alpha squared plus 2 alpha beta and let me just put a parenthesis for this and then plus beta square like this this right here is actually a purpose queer and I'm not really come to the square kind of but not really because the you should complete the square you just add a number and subtract number but here we're trying to make things happen okay so this is what we know how to factor this is what we have perhaps a to the fourth power let's just bring the apprentices look at a to the fourth power I kept you here as a square square of course that's a to a fourth power and then for four P to the fourth power I can put it here as to be square and then square that will give me that as well and what star said so far this part is the same as this but I would like to introduce the middle part of course for this parenthesis I will just bring this here namely a square and then for this parenthesis out bring this here namely 2b squared and when we have this expression this is so wonderful because we can factor it but the problem is that this is no longer the same as the original it's okay let's just go ahead and stop try this turn right away after this right here and this is pretty much what I have done in the previous video right you add something and the minus something to make things happen anyway we write this down and of course two times two we can write this yes for a squared B squared like this and of course you can verify the right hand sides the same as the left hand side and the beauty of the letters the beauty of the right answer is that this three terms together well be just a square inside and then we add to be square like this and then square that so all these three terms you become this after you factor it and the best part is that for this I can look at this as minus parenthesis with a square on the outside and of course filling the parentheses this will be to a B right and the do is that you see this right here it's a difference of two squares now and we can actually factor it okay right so you will see this right here will give us well it's this minus that times this plus that in terms of the inside so let me write it down we will have a square plus two B Square and first let me just write down minus this guy which is 2a and then we are going to multiply by a squared plus 2b squared and then plus 2 a B right here and this is so wonderful isn't it yes and as you can see in our situation a is 2019 and B years 2 to D 1009 which is really good because now we can factor it by just using this identity so let's take a look of what we end up with as you can see we have a squared minus 2 a B so let me just write it down right here a is 2019 so we have 2019 squared and then minus 2 times a and B so we have 2019 times this right here is the P so we put on 2 to the 1009 lat to P squared which is this thing squared let me just write down 2 to the 1009 and this right here is the first factor and then let me just can like box discus I kind of run out of space I bet and then we multiply with the second factor which is pretty much the same thing instead you have a plus right here instead of the - there so as you can see we actually were able to factor this into a product this number times this number and now you have to be careful you have to show that this is 9 4 to 1 and this is not equal to 1 because if one of the numbers is equal to 1 technically you are not really factoring right of course this right here it's really clear that it's greater than 1 why because everything is positive and you're adding and these are crazy numbers of course this is greater than 1 our vsd and this is kind of study trickier because you see you have a subtraction the bit in the middle right here so you might have to do some convincing work this is clear but this right you had to do some work so notice this right here is 2 times totally multiply the powers you get 2018 and of course you multiply this which you just add a power which is this is just the same as 2 to the 2019 I think this right here is bigger than that well let me just make a note as you can see it right here and here we can defy both sides by this power namely two to the 1009 lat two to the 1010 like this yeah you subtract two powers is this greater than this of course is come on I think no it's courier like two to the one thousand something of course is yeah you should be clear now okay it should be clear enough well to the twelfth power P bigger than its own of course yeah so seriously okay this right here it's bigger than this and of course you're adding that with it has to be bigger than one and it depends on how you want to phrase it but you can just clear it's bigger than what so the conclusion is that 2019 to the fourth power plus four 2020 nineteenth power is not prime and before we go I have a question for you guys when can they expression and to the fourth power plus poetry and power be a prime number be sure to leave a comment down below and let us know your answer and they get so much watching and if you guys like this video be sure to give me a like and also subscribe they get so much and as always that's it you