welcome back to my channel hi I'm mathematics happy to see you here and really interesting Olympia challenge today a to the 4 power equal to Aus 1 to the four power maybe this challenge you've ever seen but a lot of student do this common mistake and I'm going to show you this mistake here in just one minute so it will be really interesting first of all let's rewrite our challenge so we have a to the fourth power equal to a minus 1 a - 1 to the four power and a lot of students Sol this question like that they say okay we have Force Power on both side on the left on the right hand side so let's let's apply Force root for example and we have something like a equal to a minus one and one part of student tell that correct answer is something like2 but this is not a great solution to this challenge we should know a really great approach and in this video I'm going to fully explain you how can solve it correctly and step by step because we forget about Roots right here we have Force Power so we have maybe four roots right here so let's let's look at it first of all let's bring this expression from the right hand side to left hand side so we have a to the 4th power minus a - 1 to the fourth power okay so we just bring this expression from from right to left okay what we're going to do next right now let's write this a to the four power s let's write it as a squar to the Power Square okay so I hope you understand this step and this a minus 1 to the 4 power with the same logic let's write it as minus so we have let's write these brackets minus and right here we have a min-1 squar and we raise all of this to the power squ Square equal to equal to zero okay what we going to do next of course difference of squares all known formula we have for example x² - y sare we can write it as as x + y yeah someone write it like that with the plus x - y okay and let's apply this formula right here this is difference of squares so let's apply this formula in this case we have exactly the same thing we have first value square and the second expression we have we have squared so let's do that so we have a long parenthesis with the with the minus sign with the plus sign doesn't matter I start with minus sign okay so we have a square minus right here we have a - one square a -1 squar yeah and we multiply it by the same thing with a plus sign okay so we have a square plus a - one a - one square and equal to equal to zero okay so I hope you understand this step I hope you understand this uh formula right here and right now let's try to let's try to simplify this a little bit so let's do this so we know this formula this is a minus b square so a square - 2 A B minus b square okay so we have first parenthesis a square minus and this formula a square so we just rate this to the power second yeah a square - 2 a and plus and + one and of course the second parentheses so we have a * a square + a - 1 square so we have the same thing so we have a square - 2 a and + 1 and equal to equal to zero okay let's look at it real real quick so right here let's open our parenthesis with the minus sign and right here with the with the plus sign okay so let's go right here so let's do this so let's rewrite it real quick so we have a square minus a square + 2 a and minus - one okay minus one and the second parentheses we have we have right here a square + a square yeah we have a square + a square Min - 2 a and + 1 equal to equal to zero okay right now we have a square - a square we can easily cancel this so in the left parenthesis we have 2 A minus one and on the right parenthesis we have a squ + a square we have 2 a s - 2 a and + 1 which gives us which gives us the zero right now we have a product yeah right here we have a product of two parenthesis it implies that one parenthesis equal to Z so the first parenthesis is equal to Zer and the second parenthesis is equal to Z let's start with the second parenthesis we have 2 a squ - 2 a + 1 so we have 2 a s - 2 a and + 1 gives us zero right now let's find discriminant real quick everyone know about this coefficient right here we have a equal to 2 Bal to minus 2 and c equal to and c equal to 1 okay so let's find our formula let's apply let's uh put these values inside our formula so we have a second and third because right here we have a first okay equal to so we have minus B so we have Min - - 2 +us square < TK b square so - 2 square - 4 * 2 and * 1 okay - 4 A and divide all of these by 2 a in our case 2 * 2 equal to equal to 4 so as a result we have our a second insert equal to right here minus minus we have plus plus minus right here we have uh four minus we have right here looks like 8 so as a result we have 4 - 8 so square root of Min -4 and divide all of this by by four okay we can easily simplify this a little bit because our our Square root of-4 does not exist so right here we have we have complex Roots right here so let's simplify this a little bit so equal to let's continue right here okay so let's continue right here so right here we have two plus minus so let's write it so we have two I want to separate our Solutions yeah so right here we have two plus minus Square < TK of min-1 * 4 okay let's write it like that and all over all over four which gives us right here so we have a square root of -1 = to I so we have we can write it as 2 + - s < TK of -1 * s < TK of of 4 okay over four and as result right here we have I so 2 +us 2 +us squ > 4 equal to 2 so 2 +- 2 I and all over all over 4 let's divide both both uh Elements by 4 and our correct answer is 1 + - I over over two okay and this is our second and third roote but what about the first root right here we have our our equations of 2 a - 1 = to0 2 A = to 1 and a = to 1/ 1 / 2 a equal to 12 okay this is our a first this is our a second and third so let's write our full our full solution but I don't have enough space but let's do this right here okay so our answer our answer X first = to 12 X2 equal to 1 + I / 2 and xert = to 1 - I over over 2 this is our solution to our question this is a full solution to this question we find all possible all possible Roots right here to our challenge so this is our answer this is our our Olympiad Olympiad challenge so I hope you understand my explanation I hope you learn something new and I hope you you learn something new of course because this is a great tricky challenge a lot of students take Force root on both side a lot of students Sol this question like that for example imagine we have a to the 4 equal to a minus 1 to the fourth power a lot of students take fource root on both sides they say okay right here fource root so four root of a to the fourth power and equal to four root of a - 1 to the 4 power a - 1 to the 4 power okay something like that and then the these students write this question like that they say okay right here we can easily cancel this so we have a equal to right here we have the same thing equal to a minus one they write like that and from here we have we have really weird thing because right here we can't find our root this is very bad very bad solution you should put if you want to do this you should put absolute value right here something like that you you can easily find this root but not complex root you find root but what about complex root complex Roots right here so this is very tricky tricky challenge a lot of students forget about it a lot of students don't know how can we solve this Challenge and a lot of students take this first through they solve this question absolutely randomly so forget about this about this solution so let's go through the steps real quick first of all uh bring them from the right to left then difference of squares and two parenthesis first parenthesis and the second parenthesis is okay so right now you I hope you understand this uh this question this is really great challenge for us because a lot of students forget about this solution right here a lot of students don't know how can Sol it correctly a lot of students take Force root on both side which is really bad mistake in terms of mass and I hope you understand this explanation I hope you learn something new in this challenge I hope you understand why we have three roots not only one half which which gives you a lot information in terms of mass because this is really great really great challenge for us so definitely don't feel bad if you go this wrong if you need help with any of these classes I have a lot of videos so with that being said I wish you all the best in your adventures in your mathematic Adventure I hope you like math I hope you love solving challenges every day and wish you all the best see you in the next videos and if this video is helpful in anyway don't forget to like And subscribe and that definitely help me out for this particular video it helps me a lot and also thank you for your support for your support to my channel I it inspires me a lot it um indicate that you like my videos and I really want to say thank you for your for your support it really helps me a lot and see you in the next videos have a great day and take care of yourself