The Hardest Exam Question | Only 6% of students solved it correctly

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Math Beast

Math Beast

Күн бұрын

Пікірлер: 114
@gettingkilt
@gettingkilt Ай бұрын
I solved this directly, cubing then squaring twice. Just as simple and got the same answer.
@virtual-viking
@virtual-viking Ай бұрын
And unlike the shown method, you don't need to make magic guesses for which X to use.
@gvillamaruno
@gvillamaruno Ай бұрын
I think the same. I assumed the video was going to show a way to solve the squareroot of 2.
@Mike-H_UK
@Mike-H_UK 27 күн бұрын
@@gvillamaruno Well, I suppose you do get that the square root of 2 is approximately 19601/13860 as a rational number since (SQRT(2)-1)^n is approximately zero for large n (like 12)....
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
That's too great approach
@radupopescu9977
@radupopescu9977 29 күн бұрын
I would to like this: ((2)^(1/2)-1)^12=x ((2)^(1/2)-1)=x^12 solve 0.4142...=x^12 which have 2 real and 10 complex solution. Choose the positive real solution. Your way is too complicated. You course you need a calculator, but it is way easier.
@jamesatkinson12
@jamesatkinson12 15 күн бұрын
(2^0.5)^12 (1-1/2^0.5)^12. Use expansion series (1-x)^n for X
@PhilipSiu-w6q
@PhilipSiu-w6q Күн бұрын
x = sqrt (2) - 1 x2 = 3 - 2 sqrt (2) x3 = 5 sqrt (2) - 7 x6 + 14 x3 + 49 = 50 x6 = 1 - 14 x3 = 99 - 70 sqrt (2) x12 = 9801 + 9800 - 13860 sqrt (2) = 19601 - 13860 sqrt (2)
@guyhoghton399
@guyhoghton399 Ай бұрын
Let _a = √2 - 1, b = √2 + 1_ _a - b = -2, ab = 1_ _a² + b² = (a - b)² + 2 = 6_ _a⁴ + b⁴ = (a² + b²)² - 2 = 34_ _a⁸ + b⁸ = (a⁴ + b⁴)² - 2 = 34² - 2_ _a¹² + b¹² = (a⁸ + b⁸)(a⁴ + b⁴) - (ab)⁴(a⁴ + b⁴) = (34)(34² - 3) = 39202_ Let _a¹² = x + y√2, b¹² = x - y√2_ ∴ _2x = 39202_ ⇒ _x = 19601_ _a¹²b¹² = x² - 2y² = 1_ ⇒ _2y² = x² - 1 = (19600)(19602)_ ⇒ _y = ±(2²×5×7)(3²×11) = ±13860_ _a < b ⇒ _*_a¹² = 19601 - 13860√2_*
@assiya3023
@assiya3023 26 күн бұрын
شكرا تحليل جميل
@evgenijstoljarov2276
@evgenijstoljarov2276 13 күн бұрын
Actually b¹² = z - y√2 . How do you justify that z=x ?
@matthewkendall5235
@matthewkendall5235 27 күн бұрын
Once you see the trick to get an x^2 = 1 - 2x relationship - you could just use the binominal theorem to go right to the answer... it's having the insight to get to this point to break the problem down to something that is just straight algebra. Nicely done!
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
I thought this was simpler way of sorting out the problem although its little bit long... Thanks a lot for your precious feedback 🙂
@rafalablamowicz7919
@rafalablamowicz7919 15 күн бұрын
Let x=sqrt(2) - 1. Then, p:= x^2 + 2*x - 1 = 0, or, p is the minimal monic polynomial of x. Then, x^4 = -12*x + 5; x^8 = 169-408*x; and x^(12) = x^4*x^8 = 5741 - 13860*x = 19601 - 13860*sqrt(2).
@ahsgdf1
@ahsgdf1 26 күн бұрын
Nice, but I don't see an advantage in using the "trick" presented here over the direct calculation as follows: x^2 = (sqrt(2)-1)^2 = 2 - 2 sqrt(2) + 1 = 3-2 sqrt(2); x^4 = (x^2)^2 = (3-2 sqrt(2))^2 = 9 + 8 - 12 sqrt(2) = 17-12 sqrt(2); x^6 = x^2*x^4 = (3-2 sqrt(2))*(17-12 sqrt(2))= 99-70 sqrt(2), hence x^12=(x^6)^2 = (99-70 sqrt(2))^2 = 99^2+2*70^2 - 2*70*99 sqrt(2) = 19601-13860 sqrt(2) # Or, even shorter, x^12 = ((x^3)^2)^2.
@Patrik6920
@Patrik6920 25 күн бұрын
did this firs before relizing i likely even simpler use (a-b)^2=a^2 -2ab +b^2 wich gives (a^2-b^2)^3(a-b) wich can be expanded using cube rule ... [ (a-b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 ] wich gives a^6 + 3a^4b^2 + 3a^2b^4 + b^6(a-b) .. and a = √2, b = -1 -> a^3 = 2√2, b^3 = -1 a^6 = 8, b^6 = 1 -> x^2 - 39202 x + 1 = 0 OR .. 19601 - 13860 sqrt(2) ..havent bothered ..
@tungyeeso3637
@tungyeeso3637 24 күн бұрын
I've been here. I literally took it seriously the very first time I came across such circumstances. But then I began to realize they all were here to give us a pleasant time by making fun of themselves. I then began to take them lightly and was equally happy to make fun of myself by taking them seriously. I don't know about the rest of the viewers, but I sincerely thank them of their endeavor. 😮
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
If there are more than one ways to solve the problem. One can choose the way that is easy to explain and to calculate confidantly🤗
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
Bro you are right All are content creator here😌 For the means of earnings 😉 Its not such easy to prepare a solution, making video, uploading than watching it. Its your choice whether you like it... enjoy it, encourage it or criticse it. The fact is that😊
@Patrik6920
@Patrik6920 23 күн бұрын
​@@MathBeast.channel-l9i yes, but its always nice to know different ways of solving a problem, my solution may be obvious to many, but a complete mystery to others .. knowing different ways is always good .. and despice my solution, to even get ther u need some other knowledge, and sometimes longer solutions teaches more... i agree we should use shortcuts and teach them but u need to also learn why .
@matthiaswille8641
@matthiaswille8641 Ай бұрын
On a serious note, I tried to apply Pascal's triangle. Buah...I ended up using ChatGPT to spew out the solution. It works, but I must say, Pascal's triangle at n=12 is a nightmare. The provided solution is definitely more elegant.
@kuyajon
@kuyajon 12 күн бұрын
I solved it by going directly to the end of video and got the answer
@dominiquesebille1123
@dominiquesebille1123 Ай бұрын
seems to be complicated I just decomposed the 12th power into squares and that it easy peasy
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
That's too great approach 🙂
@pbierre
@pbierre Ай бұрын
What do you mean by "solving" this expression? If I poke the expression into a calculator, and crunch the numbers, have I "solved" it? Is 0.000025508902624 a valid solution?
@Tommy_007
@Tommy_007 26 күн бұрын
No. Your calculator has given an irrelevant answer.
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
Yeah Boss You are right 👍
@ACRead
@ACRead 20 күн бұрын
The calculator answer is not a mathematically exact answer, but it is far from 'irrelevant' - it is a valuable close approximation which gives us a far better understanding of the real-world magnitude of the answer. Looking at the mathematical answer would lead us to suspect the answer is of the order of a few 1000, whereas it's less than 1/1000 in reality!
@Russ--R
@Russ--R 20 күн бұрын
I can't see how the final answer in this 'solution' is any better than the original expression. It's still written in an implicit form, so is useless. Just seems like wanking off maths for the sake of it.
@jeremyhorne4075
@jeremyhorne4075 Ай бұрын
I decided to use Pasqual's triangle and do binomial expansion. It took about five minutes longer, and I did all the math without a calculator (that was fun), but I got it right. 🙂
@Nikolay_Pavlov
@Nikolay_Pavlov 20 күн бұрын
Use Pell numbers: a[0] = 0, a[1] = 1; for n > 1, a[n] = 2*a[n-1] + a[n-2] easy calculation 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860 a[11]=5741, a[12]=13860 (sqrt[2]-1)^n = (-1)^n * (a[n-1]+a[n] -a[n]*sqrt[2]) for =12 (sqrt[2]-1)^12 = 5741+13860-13860*sqrt[2] =19601-13860*sqrt[2] Harvard Yopta ;)
@MathBeast.channel-l9i
@MathBeast.channel-l9i 20 күн бұрын
Excellent approach 👍
@Nikolay_Pavlov
@Nikolay_Pavlov 20 күн бұрын
easy proof 1.It's easy to see (1-sqrt[2])^n is b[n]-a[n]*sqrt[2] 2/ we will try find b[n+1] and a{n+1] (1-sqrt(2))^(n+1) = (1-sqrt(2)*(1-sqrt(2))^n=(1-sqrt(2)* (b[n]-a[n]*sqrt[2])= (b{n]+2a[n]) - (b[n]+a[n])*sqrt[2] b[n+1]= b{n]+2a[n] a[n+1]=(b[n]+a[n]) let's express b by a b[n+1]= b{n]+a[n]+a[n] = a[n+1]+ a[n] *b[n+1]=a[n+1]+ a[n]* use this for b[n]=a[n-1]+a[n] and a[n+1]=(b[n]+a[n]) we find *a[n+1]=2a[n]+a[n-1]* another words we proof (1-sqrt[2])^n=a[n-1]+a[n] -a[n]*sqrt[2] where a[n] = 2*a[n-1] + a[n-2] for n>1 and a[0] let set to 0, a[1] to 1
@Zenadriel
@Zenadriel 29 күн бұрын
The exact result is 1 and I solved it in my head. By the way, I'm not a math genius! You just have to know the power laws and the third binomial formula! (√2-1)^12=(√2-1)^2*(√2-1)^2*(√2-1)^2*(√2-1)^2*(√2-1)^2*(√2-1)^2=1*1*1*1*1*1=1
@akaRicoSanchez
@akaRicoSanchez 29 күн бұрын
I have bad news for you: (√2-1)^2 is not the same as (√2-1)(√2+1)
@Zenadriel
@Zenadriel 28 күн бұрын
@@akaRicoSanchez Oh, I really got that wrong. The only strange thing is that I also get 1. That's probably why I didn't look so closely.
@DonVitalio
@DonVitalio 11 күн бұрын
I got recurrent formula for coefficients of (sqrt(2)-1) ^n=a_n sqrt(2) +b_n
@RealQinnMalloryu4
@RealQinnMalloryu4 Ай бұрын
D(2 ➖ 1)3^4 (2 ➖ 1)^3^2^2 (1 ➖ 1)^3^1^2 ()^3^2 (x ➖ 3x+2).
@ACRead
@ACRead 20 күн бұрын
I would like to have heard some explanation as to why you took this convoluted route to the solution. As others have said, simply expanding a cube, then squaring twice does the job as expected. The splodgy writing does not make it easy to follow either and you've made a serious error at 9:40: "as we supposed x^12 = sqrt(2) - 1".
@matthiaswille8641
@matthiaswille8641 Ай бұрын
Hmmm, if I use my calculator to actually calculate the value, I'm faster typing in (sqrt(2)-1)^12, so I truly wonder if the solution is a 'solution' :-D (well, the expression (sqrt(2)-1)^12 looks certainly more pleasant !!!
@peterkinyanjui351
@peterkinyanjui351 17 күн бұрын
Nice approach, quite enjoyable
@MathBeast.channel-l9i
@MathBeast.channel-l9i 17 күн бұрын
Thanksss
@coachlamysaint-fleur6874
@coachlamysaint-fleur6874 25 күн бұрын
At the right side you can't replace x by. √2-1. We have x^12 = √2-1 But not X
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
No. We can replace As we have assumed x= √2-1 Not x¹²=√2-1 Sorry to say you missingly got it wrong 😌
@jasonjimenez432
@jasonjimenez432 2 күн бұрын
I'm wondering if binomial expansion is applicable also.
@BubuMarimba
@BubuMarimba 16 күн бұрын
~0!
@igorrromanov
@igorrromanov 5 күн бұрын
(2-1)^6
@owenyang9543
@owenyang9543 3 күн бұрын
I think my approach is much better :)
@MathBeast.channel-l9i
@MathBeast.channel-l9i 3 күн бұрын
Alright Boss 😊
@Zbigniew-b3u
@Zbigniew-b3u Ай бұрын
do Oxfordu nie dostałbyś się z takim rozwiązaniem !!
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
🙂
@Tommy_007
@Tommy_007 26 күн бұрын
98% just like the problem and don't care how many failed.
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
😊
@SanjanaSingh-zd5oh
@SanjanaSingh-zd5oh 24 күн бұрын
Can't we just do it throught binomial expansion. I think it will a lot be easier. I maybe wrong.
@ahsgdf1
@ahsgdf1 23 күн бұрын
Just do it, and you can judge if it really was "a lot easier" :-)
@satyapriyagogula8334
@satyapriyagogula8334 29 күн бұрын
Very good calculation.
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
Glad you think so!
@masterpiecepscchannel3346
@masterpiecepscchannel3346 9 күн бұрын
Why shuld Quadratic equation?? Why lot of steps needed
@senfilatechnologies2938
@senfilatechnologies2938 Ай бұрын
what? sqrt(2)-1 = 0.41421... then this raised to the 12 power is 1/39202...
@Zbigniew-b3u
@Zbigniew-b3u Ай бұрын
😂
@jeanlucwilain7662
@jeanlucwilain7662 20 сағат бұрын
Very subtil...
@ukdavepianoman
@ukdavepianoman Ай бұрын
...as we have supposed x = sqrt(2)-1, not x^12. Otherwise nice solve.
@jackymeillon9564
@jackymeillon9564 Ай бұрын
Tout à fait d'accord
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
Thankyou Boss..that may be a typo 😔
@jesper856
@jesper856 29 күн бұрын
Ugh that loud annoying music!!!
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
Sorry 😔 I have changed in next videos You could have lower the volume 😌
@ephraimgarrett4727
@ephraimgarrett4727 29 күн бұрын
(1,4142 -1)^12 0.4142^12 I guess Oxford isn't my cup of tea. 😄😄😄
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
Using calculator 🫣
@ephraimgarrett4727
@ephraimgarrett4727 22 күн бұрын
@MathBeast.channel-l9i Even Captain Kirk cheated when faced with the Kobayashi Maru test. 😄😄😄
@MathBeast.channel-l9i
@MathBeast.channel-l9i 22 күн бұрын
@@ephraimgarrett4727 😃😃🤗
@hanooneeme8947
@hanooneeme8947 23 күн бұрын
Wrong at 10:10
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
Please review it😌
@learnspread6271
@learnspread6271 19 күн бұрын
can we use the concept of iota and iota2
@nuranichandra2177
@nuranichandra2177 Ай бұрын
Third step to last is wrong. We did not assume x^12
@ЮрийСеменов-м8и
@ЮрийСеменов-м8и 19 күн бұрын
(✓2-1)¹²=(3-2✓2)⁶= (17-12✓2)³=. Легко
@nikolllleshi5853
@nikolllleshi5853 2 күн бұрын
X¹² =([ (X)²]²)³ = .....
@Bombfun
@Bombfun 10 күн бұрын
直接入計數機吧!蠢得如此交關。如果你不用計數機,人動爆開也很易發現每爆一步的規律
@petercrem-wl9ev
@petercrem-wl9ev 27 күн бұрын
Dragged out and thus tediously boring
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
Sorry 😔
@jackn4853
@jackn4853 Ай бұрын
Drop that annoying "music" please. What a distracting racket.
@WaLiDTm-e9f
@WaLiDTm-e9f 27 күн бұрын
😂
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
Okay 😔 I will
@regfordca
@regfordca 6 күн бұрын
I just turned the music off with the magic of fingers
@abhi-kj8sm
@abhi-kj8sm 16 күн бұрын
Not satisfy 😂
@鄧貴文-v2t
@鄧貴文-v2t 27 күн бұрын
Although the answer is correct, I saw something wrong as x^12=根號2-1 at 9:44
@lcheringer100
@lcheringer100 14 күн бұрын
0.414 to the power of 12 is equal to ... 0,00002 ... !!!!!!!!!!!!!!!!!!!!!!!!!!!!! This is not your solution.
@marelloygmd
@marelloygmd 8 күн бұрын
19601-13860√2 ≈ 0.0000255 (√2-1)^12 ≈ 0.0000255
@haroldlanni8702
@haroldlanni8702 8 күн бұрын
I got the same answer as you
@ramza2779
@ramza2779 Ай бұрын
sqrt(2) ~ 19601/13860
@marcusviniciusmenossiolive1997
@marcusviniciusmenossiolive1997 25 күн бұрын
0 (zero)
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
🤔 Sorry boss its not zero
@loveYou-po9kb
@loveYou-po9kb 24 күн бұрын
最後から二行目に表記のミスがあるような😅
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
May be.😌
@dwm1943
@dwm1943 Ай бұрын
What's the point? Just boring arithmetic, and you would assume your students could do that. Or perhaps it wasn't a maths exam?
@caperider1160
@caperider1160 Ай бұрын
I doubt it was a regular maths exam. Perhaps maths olympiad warm up question. The point is, it shows one elegant way to substitute the squared term in terms of linear terms to reduce arithmetic calculations. Believe it or not, mathematicians attain satisfaction by finding alternative ways to tackle the same problem, the more elegant the better.
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
🤗
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
🤔
@Zbigniew-b3u
@Zbigniew-b3u Ай бұрын
😁
@JITENDRASINGH-it2di
@JITENDRASINGH-it2di 21 күн бұрын
Very easy question ❤
@MathBeast.channel-l9i
@MathBeast.channel-l9i 21 күн бұрын
Okay Thanks for sharing your precious feedback 🙂
@g.k.6451
@g.k.6451 17 күн бұрын
Sir,u annoy by even considering avoidable steps too.
@Zbigniew-b3u
@Zbigniew-b3u Ай бұрын
... coś tu spierdoliłeś !!! według ciebie (sqrt(2)-1)^12 to około 0,188, przecież to bzdura !!!!!!!!!
@MathBeast.channel-l9i
@MathBeast.channel-l9i 23 күн бұрын
🤔
@advancededu1
@advancededu1 20 күн бұрын
waste of time
@edeledeledel5490
@edeledeledel5490 6 күн бұрын
Pointless question really.
@С_666-р1ч
@С_666-р1ч 15 күн бұрын
И нафиг такой бред? Там практически всё легко прям в лоб.
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