Math Olympiad 3^m-2^m=65 | Math Olympiad Problems | Algebra

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In solving this math Olympiad problem, 3^m-2^m=65, Jakes uses a very unique approach to handle this exponential math challenge with easy.
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28 июн 2024

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Комментарии : 3 тыс.

@tiehoteele874 6 месяцев назад

The comment section is polluted by critics but i learned some tricks from this video.. EXCELLENT

@onlineMathsTV 6 месяцев назад

Wow!!! Thanks and we are glad you gained some values from this video tutorial sir.

@thefireyphoenix 2 месяца назад

it isnt critism.. in every maths channel you will see people doing that.. they do it for alternate solutions and writing down their own way

@tiehoteele874 Месяц назад

@@thefireyphoenix this video wasn't made for them

@Theuserisadick Месяц назад

Polluted by critics?? You're definitely not a student of maths

@najmulhussainlaskar7118 15 дней назад

I learnt something new from this video... And I wanna say something to dear sir please do you job not listen other criticism... Love you sor i am from India ( Bengali) i am also a Maths teacher 🧡🤍💚

@mitahaubica6498 9 месяцев назад

I immediately saw that 65 can be decomposed as 81-16, and conveniently 81 is 3^4 and 16 is 2^4, so matching coefficients suggests m is 4.

@BrightonMutero 7 месяцев назад

You are a genius

@nicadi2005 7 месяцев назад

@mitahaubica6498 "I immediately saw that 65 can be decomposed as 81-16" - That's not decomposition... You can find an infinity of pairs of numbers that have their difference equal to 65. The fact that you selected one such pair that also happens to be powers of the respective bases in the original problem merely indicates you have approached solving this by trial and error... *The question would be whether you can do better than finding the solution that way...*

@counterpoint9260 6 месяцев назад

that is not the right method..may work here but not al the time

@davidmajor1508 4 месяца назад

You just got lucky. It was pure luck that you used the right numbers to subtract, and that m is an integer in this case.

@danielvazquez6301 2 месяца назад

Always try some values of m to see the behaviour. m=0 or 1 or 3 or 4... I've found the solution! Obviously it is not an Olympiad problem.

@danieldavies1829 9 месяцев назад

I worked it out a bit different. My solution was simply determine what of 3 exponent would get me a number greater than 65 that would be an odd number (3^4). I then subtracted that 65 from that number (81) and I got 16 which is 2^4. In other words you can rewrite the equation in this instance as 3^m - 2^m=65 3^m - 65 = 2^m The first exponent of 3 which results in a number greater than 65 is 4 so 3^4 = 81 81-65 = 2^m 16 = 2^m 16 can be written as 2^4 m=4

@Xhopp3r 9 месяцев назад

That's exactly what I did.

@Smith_14 9 месяцев назад

m can be a negative number?

@leishajuneja2994 9 месяцев назад

You cant take any value of 'm'.consider the question is same but with a very large value(instead of 65),probably in crores,it wout take an eternity to reach that number

@leoosu 9 месяцев назад

I did the same😂

@DownhillAllTheWay 9 месяцев назад

You're assuming that m is an integer?

@f-s406 9 месяцев назад

I got ‘m=4’ by mental arithmetic. Because 3^m > 65 and 2^m < 3^m, that’s necessary. The value '65' determines that the value range of m must be less than 5 and greater than 0. When m is a positive integer, test the m=5, 4, 3, 2, 1 and finally get m=4.

@CBSE24 8 месяцев назад

I too

@tuyu6404 8 месяцев назад

Since we don't participate in the Olympics, this short answer is good. But I'm not sure if I would use this method for an Olympics. I think logarithms are simpler than the answer And have a good axiom to condense those complicated answers.❤

@lophocthienuc7345 7 месяцев назад

I dont think so. 2^m 65 why m

@gheorgheneacsu3356 4 месяца назад

Me too! 😊😊

@chrissyday67 2 месяца назад

exactly! if students are good enough to do olympiad problems then they'd know powers of 3, 3, 9, 27, 81, 243 at least and also 2 to even higher powrers, 2, 4 , 8, 16, 32 etc so very easy to work out in less than 20 seconds . However I'm used to doing mental arithmatic as I never had a calculater when I was in school

@rubikaz Год назад

There is a problem here, when you write (x+y)(x-y)=5x13 you can not deduce that x+y=13 because you do not know if x+y is a natural number. If m is an odd number, then x=3^(m/2) and y=2^(m/2) are not natural numbers. So you have prove that if m is an even number, then m=4. it is very easy to check that there is only 1 solution.

@onlineMathsTV Год назад

Noted.

@nicadi2005 6 месяцев назад

@rubikaz "So you have prove that if m is an even number" - Or, you can assume that m is even (thus making the quantities x+y and x-y (positive) integers etc.) and see whether it pays off - which it does, actually. "it is very easy to check that there is only 1 solution." - Indeed. The uniqueness of the solution is a direct consequence of the properties of the exponential function at work here...

@babetesfaye1001 4 месяца назад

❤❤❤ May I use any number Which have same numerator band denominator ? like 3/3 , 4/4 , 5/5....

@with.love.from.siberia 4 месяца назад

@@babetesfaye1001да, потому что оно равно 1

@Quasar900 4 месяца назад

@@babetesfaye1001 the function f(x) = 3^x - 2^x where x>0 is strictly growing, therefore with x=6 , f(6) > 65 so x must be less than 6, and so on trying integers until finding x=4 or m=4

@nicholastergech8525 10 месяцев назад

Similarly you can as well re-write 65 as 81-16....From there you make the bases of the two numbers to be similar with what you have on the left hand side..From there you take one of the corresponding bases and equate them together,when bases are the same powers will also be the same.

@usmanmusa8028 10 месяцев назад

This is what I actually expected from him

@TomJones-tx7pb 10 месяцев назад

If you do not notice that 16 = 2**4, you are not a computer geek.

@lgmoses3876 10 месяцев назад

I did it,in my braine.

@sfqamd 10 месяцев назад

ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-sIlRHg730CA.htmlsi=ujk2KD_xjoMGqiP5

@shivaprasadmallikarjunaiah3751 10 месяцев назад

you are not mathematically solving the problem, but doing so by trial and error. These were smaller numbers so it is easy for anyone to come to that conclusion ( becausethe solution is "visible" in the numbers in front of you). In other words, how would you solve the same problem with entirely different and larger numbers involved? ...say 2059 for instance.

@alfredomulleretxeberria4239 8 месяцев назад

I was impressed by the analytic demonstration used to figure out that m = 4. Sometimes procedures can be more interesting to follow along than just knowing the result.

@italixgaming915 8 месяцев назад

Well, you'll be impressed to see that the proof is not valid. If you suppose that m is an even number then x+y and x-y are integers but if it's an odd number then they are IRRATIONALS and you can't use 65=5*13. Plus the fact that even in the case where x+y and x-y are integers, 5*13 is not the only way to get 65, you must also look at 1*65...

@jarikosonen4079 5 месяцев назад

@@italixgaming915This looks like diophantine method used here, which would work only with the integers. In case the solution works like in above case it could prove that no other integer solutions exist... But it seems 65x1 was not checked. This could work for 3^m-2^n cases also. Variables 'm' and also 'n' are often used for integers, but not necessarily always.

@claudiohagra 4 месяца назад

You are the top!😊 You are a great teacher!!!!

@elmehdiazzouz7888 3 месяца назад

the proof is not valid, we need to proceed much more cautiously with analytic ways, i d say : some more conditions/discussions needed to be added to the video .... i gree with italixgaming the arithmetic way stays safer ..

@samuelmayna 8 месяцев назад

You can also factor 65 into 65 and 1. This gives values of a and b as 33 and 32 hence b=2^(m/2)=32, m= 10 but m will have different value for 3^(m/2)= 33. Using logs(or ln) m=(log 33/log 3)=6.365

@eliasgitau7353 7 месяцев назад

This makes sense and it is mathematically correct

@nicadi2005 7 месяцев назад

@samuelmayna "You can also factor 65 into 65 and 1." - Yes, you can, but these won't be proper factors for 65, in the sense that ANY NUMBER could be "factored" as itself and one... Also, as you've seen yourself, this breaks the consistency of the original equation by forcing the unknown to take different values simultaneously - which is obviously not possible.

@user-uy7uu1rm5u 6 месяцев назад

m is an integer

@samuelmayna 6 месяцев назад

@@nicadi2005 but it is mathematically logical.Mathematics is about thinking all cases.

@samuelmayna 6 месяцев назад

@@user-uy7uu1rm5u but my approach is sound which shows that 65 and 1 won't work but it can give different answers for some equations.

@sanmus100 Год назад

How can one just assume that x+y = 13, and x-y = 5, respectfully, as there's 65 and 1 as well. Furthermore, this wouldn't really work if 65 had many more factors, making it more complicated, opening up to a lot more possibilities.

@onlineMathsTV Год назад

Nice question @Santhosh John. There are principles and rules that govern the operations in mathematics as it is for all other sphere of life. Once you are a maths student or tutor you must get urself familiarized with these rules. They become part of you and you know what to do once you have a math challenge/problem before you. Once a mathematician sees a math problem, his head automatically runs through different means of approaching the problem for a better solution.

@thegreathussar9442 Год назад

When taking difference of squares if smaller part is 1(assuming it is the smaller part and it is) , m=2 and positive part becomes 5, not 65 therefore the result will be 5, not 65. If it has more factors, you just have to make arithmetical inferences and simplify it. That's it.

@naharmath Год назад

3^(m/2) is not necesserly an integer!

@thegreathussar9442 Год назад

@@naharmath but there is no other solution since both parts are exponential and even if m is a rational number the result wouldn't be integer( they are different primes). So this equation requires to be analyzed numerically first

@mariosantangelo9929 Год назад

Il professore ti ha risposto in maniera adeguata. Però io consiglio al professore di spiegare certe regole anche se ciò richiede qualche minuto in più. I fruitori di you tube non sono tutti matematici, ma persone desiderose di capire ed imparare.

@georiashang1120 9 месяцев назад

65=13•5=(9+4)×(9-4)=9^2-4^2=3^4-2^4 4 is the m; My high school math teacher used to tell me,the easiest way to understand an equation is to make them look the same,that is to say,we should make the brief side more complicated other than simplifing the complicated side for the most of the time.

@Amy-601 8 месяцев назад

The way I see it, 3 cubed is 27, less than 65, and 3 raised to 4 is 81. Therefore 65 is between 27 and 81. Upper bound, lower bound or range even. Now 81 would mean m is 4. So 2 should be raised to 4 also which gives us 16. 81 minus ➖ 16 is 65. So m is 4. The other ways are using log or binomial series which is overkill for smaller numbers. - Amy

@Sapped6 8 месяцев назад

A matemática é uma língua universal como a música. Parabéns, ótima técnica ❤

@user-oi3iv7oo4z Год назад

It doesn't work if m is odd. In this case (x+y) and (x-y) aren't integer and can't be assumed as 5x13.

@danielrivera2278 Год назад

If you do the analysis, and supposing m is an integer, you can conclude m is even. That's because 3^m-2^m must be congruent to 0 mod (5). If m is even you get 1 mod 5 or 4 mod 5. But m being even, you have 0 mod 5 always. Anyways, it's not proven in the video, maybe it would be amazing to have hows and whys in the video

@user-oi3iv7oo4z Год назад

@@danielrivera2278 of cause. I mean that "trick" in the video is not universal.

@Change_Verification Год назад

@@danielrivera2278 and who even said that m must be an integer ?

@danielrivera2278 Год назад

@@Change_Verification exactly. I also think like that, that's because assuming integer was the first thing I said

@Eismann1 9 месяцев назад

Yes, he didn't do the preliminary work. But this is an important step to make tricks like this one work.

@Shirlippe Год назад

Hi there! Here is an alternative solution. The difference of the powers on the rhs increases with m, and it is already greater than 65 for m=5. So m must be less than 5. Assuming m is an integer, and noticing that m=3 doesn't work, the only possible solution is 4.

@onlineMathsTV Год назад

Ya boss. You are very very correct and I like you approach sir. This is an indication that you are a master in this area. Respect sir...👍👍👍

@ronitmahawar1193 Год назад

but this only works if m is integer which we dont know it would be

@romanyukvictor Год назад

@@ronitmahawar1193 Exactly! To solve this you have to prove that the equation has single solution. Just shift 2^m to the right side and divide the entire equation on 2^m. As a result you will get the increasing function of m on the left (3/2)^m and decreasing function on the right (1+65/2^m) which can intersect only once. So there is only one root. The only way to find the root is to enumerate the integers and find out m=4.

@ifomichev Год назад

@@ronitmahawar1193 the solution proposed by the author of the video also works only for integers, because it relies on factorization of 65

@jccamargo99 Год назад

For this reason many people don't like math.

@user-qo6ni5sm5p 9 месяцев назад

Solução genial. Parabéns. Muy interesante, hoy aprendí un buen método, gracias.

@GautamKumar-wx3sm 5 месяцев назад

Couldn't have thought of this approach. Thanks for this.

@elio9008 Год назад

If you guess the solution, m=4 and present the equality as 3^m - 81 = 2^m - 16 , then it would be easy to prove that both functions (on the right and on the left) increase and therefore their graphs have only one intersection.

@ca1498 10 месяцев назад

You need more than that. Two increasing functions can intertwine and cross each other all the time. But if they have one intersection, and after that one of them grows faster than the other all the time, then it follows that they won't intersect again. It's like two cars racing. They both increase their distance from the start all the time, but they could swap places many times during the race--unless one of them is always faster than the other after the point in which they were even with each other.

@ca1498 10 месяцев назад

@@reginaldocalvo4361Each of the two sides can be a function e.g. y = 3^x - 81. The solution of the equation 'side 1' = 'side 2' is a number x (or m) where the two functions give the same y for the same x. You first find, by guessing, one x (or m) for which the two functions have the same y, which would mean that the particular m is one solution to the equation of function 1 = function 2 for some x. You then show that each of the functions only grows, and that the difference between the two y-s for each x (which is a function of x as well) also only grows. Therefore there can be only one value of x for which the difference is 0, so only one m (the one we already guessed) is the solution to the equation where one of the functions has the same y as the other for a given x. If I were participating in this Olympiad and solving this problem, I would have guessed 4, and then I would have argued that the first side 3^m... grows much faster than the other side 2^m for each subsequent m. I wouldn't be using derivatives, as I did not know any calculus in high school. But if they ask about integer solutions, I would talk about growth of y with respect to changes in m by +1. And if they did not limit it to whole numbers, I would probably still try to talk about slope of the graphs of the functions and hope to make an acceptable argument, as I don't see how you can analyze these functions without using calculus.

@user-xl3mg3om7s 9 месяцев назад

@@reginaldocalvo4361 3^m-81= f(m) this can be considered as a function depending on m 2^m-16= g(m) also could be considered as function of m.

@user-xl3mg3om7s 9 месяцев назад

@@ca1498 you are right, but When you see this function it is easy to notice that the growing path or direction is already know but partially

@Vitzyk 9 месяцев назад

Consequence is false.Contra-example x and x^3. Both increase but have 2 intersections

@therichcircle.8819 Год назад

You tried here tutor Jakes. I have learnt something here. Just keeping on running this channel. More grace, love from Port Harcourt ❤.

@onlineMathsTV Год назад

It is our pleasure to serve you sir. Thanks for watching

@teamredx1pro952 3 месяца назад

Thank you for reminding me that you can't believe everything on the internet. Props to you i almost believed it untill i tested it with calculator. BRAVO YOU SHOULD WIN AN OSCAR

@fisicamatematicasprofewilliam 5 месяцев назад

profe en olimpiadas de las matematicas se aprende mucho. y con usted bastante. felicitaciones

@onlineMathsTV 5 месяцев назад

Thanks a million sir, we appreciate this comment my good friend.

@user-pd7js7cy9m Год назад

It can decide differently. The function (1) f(x)=3^x-2^x -is increasing . { for x>0 (2) 3^x>2^x ; x10 . (3) f(x2)-f(x1)= ……..=3^x1*[3^(x2-x1)-1]-2^x1*[2^(x2-x1)-1 ]> 2^x1*[2^(x2-x1) -1 ]-[ “--“ ]=0 ; (3) f(x2)-f(x1)>0 !!!!! So , it takes all its meanings once a time. f(4)=3^4-2^4=65 . So , x=4 - is the only root of equation ! Respectfully , Lidiy

@onlineMathsTV Год назад

The elders in our mist are highly learned. Love your detailed explanation sir. Thanks for finding our time to watch our content and commenting even at this age of yours sir. Much respect and we All @onlinemathstv love you dearly...💖💖💕💕

@Lernen-mit-Rudi Год назад

There is no X! 😂but however, good job!

@andreasandre4756 11 месяцев назад

Pay attention that m=constant not variable, so M must be grater then 1 otherwise solution will not be true because ln1=0 or 3-2=1 which is not equal 65. So M>1 and could be anything. So if it is not an integer number? So if it is not equal 65 but 63.5 - ?

@ADSemenov_ru 10 месяцев назад

You took the words right out of my mouth. :)

@Marat7973 10 месяцев назад

Здравствуйте,Лидий! Не ожидал Вас здесь увидеть)

@johnpalagye7036 Год назад

Love your video! I got lost half way and I did not know that you could just square the exponents and make them equal

@onlineMathsTV Год назад

We are glad you love what is happening here. We promise to give more educative contents in the area of mathematics with the help of God. Love you.....💖💖💕💕

@karlm9584 7 месяцев назад

I like this too. I have done it with roots before. Roots are just indices but I wouldn't have thought of doing it.

@dannychown2593 Месяц назад

Your work opens up the horizon of my mind how to approach this type of question. Thank you.

@marcio5457 9 месяцев назад

Solução genial. Parabéns

@airtonreis2675 11 месяцев назад

Parabéns. Não sei falar nada em inglês e mesmo assim consegui aprender com sua aula. Até eu estou surpreso de ter assistido sua aula até o final sem saber se iria entender o seu modo de esnsinar. A matemática pode ser universal, mas o jeito de ensinar é fundamental.

@onlineMathsTV 11 месяцев назад

@Airtonreis, we want to sincerely say you are such a wonderful person and thanks a million for watching our contents despite the language barrier. We all @OnlineMathstv deeply cherish and love you from the depth of our hearts sir. ❤️❤️❤️💖💖💖💕💕💕🙋🙋🙋

@airtonreis2675 11 месяцев назад

@@onlineMathsTV 🤝

@MATHSHEADMASTER 9 месяцев назад

You are welcome to subscribe to our channel for more interesting math problems.

@pedrooo13 5 месяцев назад

Brasil tá em todo lugar não tem jeitoooooo

@RoderickEtheria Год назад

Solved 3^m-2^m=65 by just thinking about the first whole number power of 3 above 65.

@dilphek Год назад

It is not about finding it. Olympiad is a school competition teaching kids to solve these problems mathematically

@bleh-zj1hy 8 месяцев назад

@@dilphekyou guys got a different Olympiad or something? Here Olympiads (for the kids, totally different type from the subjective ones) are mcqs and the subjective ones are like 3 questions in 3 hrs and if you're able to do even 1 you're qualified (you can imagine the toughness so it's really not for the kids)

@jamesclerkmaxwell8545 9 месяцев назад

Parabéns. Resolução muito boa.

@user-rz7ym5ts1e 9 месяцев назад

I love how the way you work out the question..❤❤

@jaimeduncan6167 Год назад

Another option: the solution is pretty clear, it's a small number one can calculate with the mind. Then it reduces to show that their solution is unique. One can use calculus to show that the analog continuos function is monotone for x>4 and be done with it, or use induction to show that it grows on the integers for n>4.

@onlineMathsTV Год назад

Your approach is superb sir. I find it very fascinating and I will try it out in subsequent math challenges. Thanks for sharing this nice and wonderful procedure with OnlinemathsTV. You the boss and much respect boss...👍👍👍

@GirGir183 9 месяцев назад

I think he used a simplified equation for the demonstration. When it's not so simple and the numbers are much larger, then this method can be used as well.

@ravirajshelar250 9 месяцев назад

I did it in a different way. Write 3 as 2+1 and perform binomial expansion, so the 2^m cancels and subtract one from both sides. We have 64 on one side and some series on other side. Notice that the series 2 + 2^2 +.....+ 2^m will definitely be smaller than series on left which is equal to 64. So make this G.P. sum less than 64, you will get that m should be less than 5, and once you have known this, you have proved that you just need to find a solutions less than 5 and those will be the only solutions. Only m=4 works out.

@albajasadur2694 8 месяцев назад

a good method to find the range of m and it makes sensible checking easier with limited number of m

@theboss73104 2 месяца назад

Yeah

@karlm9584 7 месяцев назад

I like how you used indices to get past that first section. I've never really been good at spotting where to use substitution, the section when you brought in x and y

@onlineMathsTV 7 месяцев назад

Hahahaha....thanks a bunch my good friend and thanks for watching our contents consistently. We all here love deeply....💖💖💕💕😍😍

@brunodelenclos6035 4 дня назад

Belle démonstration.Merci beaucoup..

@franciscodeassisbrandaobra898 Год назад

exercicio maravilhoso🥰🥰❤❤❤❤❤❤

@onlineMathsTV Год назад

Thanks a millions sir, we love you ❤️❤️💖💖💕💕😍😍

@Psykolord1989 Год назад

Before watching: Alright, so, we are looking at exponential functions. 3^m - 2^m = 65. First, we can rule out m=1 and anything below; the difference between those would be smaller than between 3 and 2, and thus much lower than 65. Next, notice that we are dealing with an integer on the right. This heavily implies (but *does not necessarily guarantee* ) that 3^m and 2^m are both integers as well. If both are integers, then m must also be an integer. So we should start with integers. (If we were dealing with a mixed number instead, this would be much more complex; as it stands, we can just plug in integers and see which one works). Let's start with the first x that gives us 3^m >65, namely 4. 3^4 = 9^2 = 81, and 2^4 = 4^2 = 16 So for m= 4, we have 81-16 = 65. Fortunately for us, this checks out, and thus we have our answer of *m = 4*

@onlineMathsTV Год назад

Thanks for this thorough explanation. You the best. 👍👍

@RoderickEtheria Год назад

Negative m gets fractions.

@Psykolord1989 Год назад

@@RoderickEtheria Yes, you are correct, and I see I made a typo in the "so for m=4 we have..." section by putting a negative in front of the 4. Fixed now. I don't imagine it was a huge problem since in the section right above it,and at the very end of that section, I used 4, but it still may have confused some people.

@danielmelo5609 8 месяцев назад

Excellent explanation. It seemed very complicated, but it turned out to be easier than expected. A hug

@user-og4vc7ey2y 4 месяца назад

I'm impressed with this explicit methodology 😊(12:16am)

@umeshkhetan Год назад

If a.b=65, a and b can have infinite values. So, the tutor has just one answer where multiple answers are possible.

@onlineMathsTV Год назад

Yes it has multiple answer but for the sake of this tutorial we restricted ourselves to this solution sir. Thanks for this observation. Much love....💕💕👍👍

@mustaphaolunrebi8100 9 месяцев назад

I think you add that, where m is an integer. It makes it complete. The only integer factors 65 has are 5 and 13. Nice solution 🎉

@ivandonchev474 9 месяцев назад

Sorry but this video is the most useless shit I have seen. You solved it by guessing and overcomplicated massively

@speedsterh 8 месяцев назад

@@mustaphaolunrebi8100 No, 65 has 4 factors: 1, 5, 13, 65. The equation with 2 other factors should be explored for completeness

@mdmahin7299 Год назад

Very well solution but the rules are very lengthy or Expensive ☺️ So if we assume the value of "m" here from 1-4 We are easily getting the value of m Such as Let , m=1,2,3,4,... ♾️ and now, 3^1-2^1=1≠65 again 3^2-2^2=5≠65 And now if we let, m=4 then 3^4-2^4=65=65 So we can easily get m=4😊

@onlineMathsTV Год назад

Bravo👍👍👍

@WhiNikkatherealone 10 месяцев назад

i mean hit and trial is always the last resort

@Saibatou 7 месяцев назад

You are just great! A very good approach.

@KidusYared-ox6qb 9 месяцев назад

Thank you so much. That's the easy way to solve it❤👍

@owlsschoolofmath9732 Год назад

Great! Its a fun problem. I did something kind of similar with difference of 2 squares.

@onlineMathsTV Год назад

Nice, you the best. Kudos 👍👍

@Quasar900 10 месяцев назад

the function f(x) = 3^x - 2^x where x>0 is strictly growing, there for if x=6 f(x) > 65 so x must be

@onbored9627 5 месяцев назад

All I did was think whats the first interger power of 3 that goes past 65, 3 is 27 so the answer is 4. Then just test. and it gave right answer. simple.

@Quasar900 5 месяцев назад

@@onbored9627 Where are you from ? Cause how did you know I was still alive after 4 months ? 🙂Here is a secret about me : I 've never studied Mathematics in English !

@onbored9627 5 месяцев назад

@@Quasar900 Ah, I apologize I only speak English. I'm from the USA. I wasn't trying to take away from your explanation I should've been more clear on that, I just thought yours was so good I didn't even need to say. Figuring out it's strictly growing is clever as hell. I didn't even think of that. I meant simple, as in, my answer was simply a guess really and it worked.

@Quasar900 5 месяцев назад

@@onbored9627 Oh please Sir , no need to Apologise , The fact that I've never studied math in English doesn't mean I don't know English 🙂 It's just I'm not that familar with English terms in math ! But Thank God I do speak and read : French, English, Arabic, Spanish + some Japanese ! I did study math in French (after high school) and Arabic (until high school) , but that waaaas 21 years ago ! What class are you in ? I hope you're safe from those ongoing blizzard storms ! Greetings From Morocco and Free Palestine 🙂

@Quasar900 5 месяцев назад

@@onbored9627 I think you do know these techniques involving the continuity of a function, the intermediate values etc.. to solve equations ! For example : solving in IR set : Arctan(x+1)+Arctan(x-1)=Pi/4

@chrisw4562 9 месяцев назад

Nice job! I got the answer by trial and error, but this way of getting a deterministic solution is really cool.

@italixgaming915 8 месяцев назад

Actually this proof is wrong. If m is an even number then x+y and x-y are integers but if m is an even number then they are IRRATIONALS and the unicity of the prime factors works only with integers. Euler did the same kind of mistake when he tried to demonstrate the Fermat theorem (that time he used complex numbers).

@ludmilak9396 2 месяца назад

Очень грамотное изложение, и очень удобно следить на доске. Однозначно плюс!🎉

@pwmiles56 Год назад

In a slightly fancier approach we can make a recursion 2(3^m - 2^m) + 3^m = (2+1)3^m - 2^(m+1) = 3^(m+1) - 2^(m+1) Put a(m) = 3^m - 2^m a1 = 3^1 - 2^1 = 1 a2 = 2 a1 + 3^1 = 5 a3 = 2 a2 + 3^2 = 19 a4 = 2 a3 + 3^3 = 65, done

@onlineMathsTV Год назад

Wow!!! This approach is impressive but a bit obscure sir.

@vejayashanker 9 месяцев назад

can u explain what u mean by obscure pls😂

@soltanchalkarow905 6 месяцев назад

can you help me sir? if ab+bc+ca=1 prove: sqrt(a + (1/a)) + sqrt(b + (1/b)) + sqrt(c + (1/c)) >= 2( sqrt(a) +sqrt(b) + sqrt(c) )

@DrMikeE100 9 месяцев назад

It took a fraction of a minute to recognize m = 4, but as a mathematician, I did find this interesting. (Dr. Mike Ecker)

@JaroGoraJ 9 месяцев назад

I think because it was easy question

@sevgininbahcesi5222 4 месяца назад

Perfect.Thank you sir.

@CleopatraNadesan-kn3jg 9 месяцев назад

Excellent tutorial. Thank you very much

@FractAlkemist Год назад

I have been learning Genetic Algorithms in Python; they are good for problems like this. The value I get for 'm' is 3.97, 3.99, 4.00, etc.; different each time as there is a random element for the convergence. A little intuition is also required; If you plug 4.0 into the equation you get correct 65. Program run time ~5 seconds.

@onlineMathsTV Год назад

Wow!!! Nice sir.

@pierrecurie Год назад

If you're going the numerical route, bisection search is much faster.

@mistertwister1015 Год назад

Blunt enumeration of roots is not always the best solution)

@victorfildshtein Год назад

Hello. I program in PureBasic. I made this task by binary search method, 30 iterations, result 3.9999999991, precision 0.0000000047. Time is almost instantaneous.

@wilsonuche9389 10 месяцев назад

You need to review the Python solutions cos only 4 is an exsct solution. 3.97 is far from it, 3.99 is just an approximate

@mathadict1 10 месяцев назад

Keep up the great work man happy to see somoen in our beloved African continent devoting a portion of their time into this much love from Morocco 🇲🇦 MA

@onlineMathsTV 10 месяцев назад

Much appreciated sir. Thanks for watching our contents and the encouragement sir. Much love from everyone @OnlinemathsTV to you sir 💖💖❤️❤️💕💕💕

@Quasar900 10 месяцев назад

@@onlineMathsTV the function f(x) = 3^x - 2^x where x>0 is strictly growing, there for if x=6 f(x) > 65 so x must be

@Quasar900 10 месяцев назад

the function f(x) = 3^x - 2^x where x>0 is strictly growing, there for if x=6 f(x) > 65 so x must be

@zulfqarali2994 2 месяца назад

Gentleman I appreciate your work.

@jamesmichuki5804 6 месяцев назад

Thanks teacher. Today I've learnt something very new in maths. Am really surprised....

@Toxa_Azimov Год назад

Слева возрастающая функция при m>0 ( можно найти производную и убедиться ), справа постоянная функция, значит у них может существовать только одна точка пересечения, методом оценки m=4

@onlineMathsTV Год назад

Bravo 👍👍👍 You the best sir.

@onlineMathsTV Год назад

Bravo 👍👍👍 You the best sir.

@PlumbuM871 6 месяцев назад

Чисто случайно подставил вместо m 4, и всё сошлось! Везёт мне

@AndrewUnruh 9 месяцев назад

Clever solution! One thing I did not get is how you knew that m was a positive integer - or was this just an assumption that happened to work? For example, if the problem had been 4^m - 3^m = 65, m would be approximately 3.36.

@charleskaruru481 8 месяцев назад

m can never be a negative otherwise we wont have 65but fraction

@user-my1vs2ep5c 14 дней назад

Thank you very much! Без перевода мне всë так понятно было👍

@gabrielschiteanu4963 2 месяца назад

Rewrite the equation as 3^x = 65 + 2^x. We can safely divide by 3^x and then we have that 1 = 65 * (1/3)^x + (2/3)^x. Because the function on the R.H.S. is strictly decreasing, being the sum of 2 other strictly decreasing functions, it means that f(x) = 1 has one solution at max. We notice that x=4 checks, so that is our only solution.

@onlineMathsTV Месяц назад

Bravo!!! You the best sir. Maximum respect sir 🙏🙏🙏

@michaelsidorov5508 Год назад

Остроумно и красиво! Как и вся математика.

@onlineMathsTV Год назад

Thanks for this wonderful comment sir. Love you....💕💕

@user-nv9rw7nh5w Год назад

Очень, очень. Надо же как можно. Удивительно!

@user-qt9qo3ez9x 10 месяцев назад

Сначала поделить обе части на 2^m. Тогда слева будет возрастающая функция, справа убывающая. Тогда уравнение имеет не более одного корня. Подобрать корень не сложно. 9 класс, ничего сложного. За проведенное решение минус: нигде не доказано отсутствие других решений, переход к системе ничего не обосновывает.

@michaellockett4044 2 месяца назад

Difference of squares into u-substitution. Excellent methodolgy.

@onlineMathsTV Месяц назад

Thanks a bunch sir.

@xuanhungnguyen7667 3 месяца назад

Chính xác.anh biến đổi 65=5×13.đây mới là mấu chốt của bài.và tư duy hàm số mũ...tuyệt.

@88kgs Год назад

Sir, we can also do this by hit and trial method, assuming different values for m=1,2,3,4.... But your way was also very nice 👍👍. Thank you for this video sir🙏

@onlineMathsTV Год назад

Yes, but that will only work if the working process is not in the examination but this is needed when the examiner wants you to show your procedure step by step...👍👍👍

@gregfarnham5651 Год назад

Yes, trial and error could work if we assume m is a positive integer. I don't believe that was a given, however.

@ralfimuller8948 Год назад

@@gregfarnham5651 The solution in the video also made use of the assumption that m is an integer. Otherwise the factorization of 65 would not be unique. Actually, the hit and trial method should be entirely ok.

@gregfarnham5651 Год назад

@@ralfimuller8948 Agree. Thank you.

@danielrivera2278 Год назад

Also, by trial and error you can't prove that's the obly answer

@AntoninaKa-es8tv Год назад

Очень интересное решение! Спасибо!

@onlineMathsTV Год назад

You are welcome always even as we look forward to seeking better ways of serving you better in our services to you on this channel. We at Onlinemathstv love you without reservation sir....💕💕💕

@marcofidelpenaavila2129 4 месяца назад

Extraordinario , thanks so much.

@jwilson4163 7 месяцев назад

Great lesson! Thanks. (From São Paulo/BR)😃

@paulmiddletonphotography4368 Год назад

Hello Online Maths TV, I really like your approach with this and how your deliver the proof. It is elegant and uses several skills. Your pace of delivery is really good too. As a suggestion for completeness in your proof, can you please include a determination table of which factors of 65 and their order are valid for consideration for equating to (x+y).(x-y). A table similar to; (x+y) . (x-y) | x | y | m from x=3^(m/2) | m from y=2^(m/2) | 1 . 65 65 . 1 13 . 5 5 . 13 If you calculate x and y and then m from x=3^(m/2) and from y=2^(m/2) for each of these arrangements, only (x+y) . (x-y) = 13.5 provides valid and consistent values for m. So 3 out the 4 arrangements can be mathematically eliminated. This would verify that the only valid arrangement and values of the factors is 13.5. I hope this helps. Cheers, Paul.

@onlineMathsTV Год назад

What a wonderful guide @Paul Middleton. Very very helpful sir. Respect boss. ❤️❤️❤️👍👍👍

@onlineMathsTV Год назад

Am compared to drop another comment in respond to your first comment sir. Am speechless and overwhelmed by your systematic construction of this comment. It is so encouraging. This is because you appreciated my little effort in the mist of professors and further suggested a wonderful/a great approach to me on solving this same problem or similar problem in my subsequent videos. Sir, with all humility we are glad to meet with you and have you here. Much love sir....💖💖💖💕💕💕❤❤❤

@paulmiddletonphotography4368 Год назад

@@onlineMathsTV You are very welcome indeed.

@paulmiddletonphotography4368 Год назад

@@onlineMathsTV Thank you for your very kind words of appreciation. It certainly is my pleasure to help you and I am really pleased that my suggestion will assist you and into the future with different proofs. May you increase your subscribers, students and people interested in learning your very clear, easy to follow and forthright teaching of maths. All the best to you for the future.

@mathtv3982 Год назад

First you must show that m is even positive integer

@onlineMathsTV Год назад

Noted sir.

@bdh9202 4 месяца назад

You worked hard, I found the result by giving the value of m to 4 in 10 seconds, but it is important how the solution is, but I still think it is too long. Greetings from Turkey.

@knguyen3348 5 месяцев назад

You are the best. Thanks

@divonsirlopes5409 Год назад

There is a faster workaround, with the assumption that the number m is integer. The term on the right is less than the term on the left. For simplicity, we can assume that the term on the right is zero. This results in: 3^m = 65. m is greater than 3, because 3^3 = 27. m can be 4 because 3^4 = 81. Let's test m = 4: 3^4 - 2^4 = 65 81 - 16 = 65 65 = 65

@onlineMathsTV Год назад

Wow!!! This is fantastic. I love this approach sir. We have gained some values from this procedure sir. Thanks for dropping this sir. Respect.....👍👍👍 Much love....💕💕💖💖❤️❤️

@divonsirlopes5409 11 месяцев назад

There is a faster workaround, with the assumption that the number m is integer. The term on the right is less than the term on the left. For simplicity, we can assume that the term on the right is zero. This results in: 3^m = 65 m = log(65)/log(3) = 3,8 m is greater than 3,8 Let's test m = 4 3^4 - 2^4 = 65 81 - 16 = 65 65 = 65

@MONSTER2013 9 месяцев назад

Base on which level you’re at. This question and video are made for yr 10-11? So he gave the solution at that grade. Above yr 12 can use other tools as log/ ln skilfully to solve it.

@divonsirlopes5409 9 месяцев назад

Thanks for the info.

@foudilbenouci482 8 месяцев назад

^you found one solution doesn t mean you found all solutions

@Curufin1984 Год назад

This is so overly complicated.. Just check low values of m and find that m = 4 works.. Then use a simple analysis tool to show uniqueness of the solution e.g. by showing that function f(m) = 3^m - 2^m is strictly increasing for m>=1. Additionally your solution contains errors and missing steps: 1. If you do the trick with 3^(m/2)^2 - 2^(m/2)^2 = 65 then you presuppose that m is even, because if m is odd then 3^(m/2) is not natural anymore, so you cannot use the natural divisors of 65 in the following steps anymore. 2. Even if that approach worked (because you somehow proved that m must be even): After you rewrite the equation as x^2 - y^2=65, then you have to consider *two* pairs of solutions 65 = 65 * 1 and 65 = 5 * 13. Long story short, lots of mistakes in your video unfortunately.

@onlineMathsTV Год назад

Thanks for this keen observation and I really appreciate this comment sir. Noted. I will do more detailed work on subsequent videos. You the best and much love for this detailed comment....💕💕💕

@ivandonchev474 9 месяцев назад

Yes this video is shit full of errors. The fact that it has so many errors and is hugely overcomplicated at the same time makes it complete and utter shit

@mikeblings1504 4 месяца назад

As a math teacher, this is a plus to me. You are amazing 👏

@onlineMathsTV 4 месяца назад

Thanks a million sir.

@rajendrasinghbisht2628 3 месяца назад

Nice solution. It makes so easy, the way you explained.

@BrukFikru 7 месяцев назад

Nice one ...... good teacher ...... stay teaching stay doing more math tricks ...... i liked it

@Luis-zj4dv Месяц назад

Genial, muchas gracias!

@marcelo372 2 месяца назад

Excelente. Thank you.

@clownphabetstrongwoman7305 4 месяца назад

Maybe it's a bad argument but I would say for a^m - b^m = X, and a, b, m belong to N, a^m> X > a^m-1. Here 3^m>65>3^m-1 81>65>27 => m=4.

@dinlendiricidrtv Месяц назад

Thank you very much my dear friend

@abdussalammondol6234 9 месяцев назад

Many many thanks.❤️💚❤️💚

@composer7325 8 месяцев назад

Excellent, thank you.

@tejendramohenbaisya8529 9 месяцев назад

Problem solved in a very intelligent way.

@TheMightyOdin 9 месяцев назад

I did this in my head in less than a minute. 3,9,27,81(bigger than 65) 2,4,8,16( subtract 16 from 81)…. Works.

@daintydawn2508 3 месяца назад

Yeah, it's a bit nd trial method. But if the value of m had been bigger then, it wouldn't work

@nnaammuuss 9 месяцев назад

m≤2 makes it too small and don't work. So, m must be even, as 3^m ≡ 3 (mod 4) for m odd, but 65 ≡ 1 (mod 4). On the other hand, 3^m - 2^m = 3^{m-1} + ... ≥ 3^{m-1} and is strictly increasing. Since, 3⁴ = 81 is already too big, we must have m

@evasolovyova 9 месяцев назад

Отличное решение, всё ясно и понятно! Как хорошо, что математика у всех одна, математика сближает людей. Мы понимаем друг друга, говоря на разных языках!

@antoniogomesfigueiredo7835 9 месяцев назад

Gostei muito da resolução dessa questão. Não entendi suas palavras, pois, sou português, todavia, a resolução da equação entendi perfeitamente. OK? Obrigado. Thank you very much.

@forgottenlegacy5929 2 месяца назад

Wonderfully explained, very informative. But it’s sometimes more convenient to use the easy method

@math_qz_2 6 месяцев назад

Very instructive task

@georgesadler7830 8 месяцев назад

Thank you for the video.

@user-ne7pu8ib7y 5 месяцев назад

3ᵐ-2ᵐ=65 y=3ˣ-2ˣ y=65 если построить оба эти графика, то будет видно, что уравнение имеет единственное решение, поэтому можно попробовать подобрать корень подбором: х=1: 3¹-2¹=1; 1≠65 х=2; 3²-2²=5; 5≠65 х=3; 3³-2³=19; 19≠65 х=4; 3⁴-2⁴=65; 65=65 х=4 корень m=4

@user-tw4fp1jg8g 5 месяцев назад

i really learn a SOME TRICK from this thankss

@charleskaruru481 8 месяцев назад

in maths olympiad time is very critical...what is important is the answer...so the best way is to solve maths olympiad is to have very basic maths then the rest is analysing to get the pattern...so what i did is just look at indecies of 3 that have the last number such that if we subract an indecies of 2 and get 65

@user-zu9lw3zb4d 7 месяцев назад

Mathematics is all abt observation and understanding thee pattern thx a ton Buddy❤

@onlineMathsTV 7 месяцев назад

Thanks a million for watching and dropping this wonderfully encouraging comment sir. The true is that mathematics is all about finding a solutions to problems/challenges. We really appreciate the fact that you watch and dropped this comment to clarify some debts in the minds of so many viewers and subscribers here. Much love from all of us @OnlinemathsTV....💕💕💕💖💖💖❤️❤️❤️