Japanese | A Nice Radical Math Simplification | Math Olympiad
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That algebraic method isn’t a method at all. You can’t set off on a series of substitutions in the hope that you might end up with an easily recognizable perfect square. The “problem” has been constructed from the “solution”.
I disagree. Once that it is recognized that 64^2+49^2+15^4 has the form x^4+y^4+(x+y)^4 the result follows from straightforward algebraic transformations. No guessing involved. Of course, in the absence of this insight, it is also not difficult to use basic arithmetic to obtain the result 169.
@@YAWTon You are right, but my point is that the “method” outlined does not mention that insight. The double substitution is totally unnecessary. (Do you agree?) You either notice that you can factorise, in which case you should (and you did - as did I when I saw this) or you go through the arithmetic.
@@YAWTon It’s over 50 years since I did any maths, but the substitutions vaguely recalled the solution to the general quartic. Is that how the “problem” was conceived, do you think?
@@richardatkinson4710 I agree that his presentation was bad for the reasons that tou mentioned. Too many intermediary steps. Apparently the problem suggests a "solution" that is invalid, but leads to the correct value (based on the assumption that the squareroot of a sum equals the sum of the squareroots of the terms of the sum).
It is a math olympiad problem !
Of course, the problem has been built from the solution : it is meant to be solved this way.
This is how the math olympiad jury makes to find the best students : by their abylity to find the right substitution at the right moment of the calculus and their ability to lead such a huge calculus until its end !
In real life maths, you hardly find such calculus but in a maths competition it's classic that people have to use such tricks
This is Candido's identity in disguise, [x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4]. Candido used the Fibonacci numbers in his original publication.
Haven't been doing this for more than thirty years ... and never missed it. 🙂
I guess calculating it directly would take less time.
You are absolutely right, it is around eight times faster.
Agree. It seems that many people doesn't knows the basics, or how make some basic arithmetic operations (using elementary knowledge, and a pencil and a sheet of paper if necessary)
@@eduardofs6620 and inclusive philosophy requires everyone to prove slowly, in small steps, that 1+1=2 😀
The fourth root would have allowed for an extra simplification, giving 13 as the final result 😊
Why not just long multiply, add, divide by 2 and then find the square root? Much simpler.
@@udayabose2357 because it doesn't work
I have much better solution. Everybody knows that 64^2=4096. 49^2=2401 15^4=225^2=50625 Half sum of it is 28561, which is square of 169, also known fact.
"Everybody knows that..." - tell us that you cheated and used a calculator without telling us that you cheated and used a calculator. 🤡
@@Swannerator Author also used calculator to find square roots of 64 and 49. It is not easier , than 64^2 and 49^2!
@@Swannerator no need in calculator, that numbers are small (easy to multiply in mind), and even if you not remember 169^2 you can just calculate it on paper.. every one knows
@@anatolykatyshev9388everyone knows sqrt of 64 and 49 💀
Concur, way much better than fiddling all that ↑↓←→abxy口O∆ around 🤦🏻♂️
I don’t care if there’s a faster way to find the answer, as someone not versed in deep mathematic equations I found it interesting.
Don't need to make any x or ys, after the first = you have (8+7)^4, that is 8^4+7^4+2x56^4. So now you easily divide all by 2, end up with 8^4+7^4+56^2 all under square root, do the root, done
169 is 13^2, so is there further simplification to be done at the 8^2 + 7^2 + 8 x 7 stage?
Прекрасная задача и великолепное объяснение ее решения. Мне 72 года и моя профессия не связана с математикой, но мне все понятно и интересно. СПАСИБО
There is no need to open the paranthesis of 4th power. You could do the second substition before that.
Excellent. Thank you.
Vamos a resolver el problema paso a paso.Primero, definimos la operación:[ \sqrt{\frac{64^2 + 49^2 + 15^4}{2}} ]Calculamos cada potencia:[ 64^2 = 64 \times 64 = 4096 ][ 49^2 = 49 \times 49 = 2401 ][ 15^4 = 15 \times 15 \times 15 \times 15 = (15^2)^2 = 225^2 = 50625 ]Sumamos los resultados obtenidos:[ 4096 + 2401 + 50625 = 57122 ]Dividimos la suma por 2:[ \frac{57122}{2} = 28561 ]Calculamos la raíz cuadrada del resultado:[ \sqrt{28561} = 169 ]Entonces, la respuesta es:[ \boxed{169} ]
Nice
Thank you very much!
Nice. Elegant.
Elegant? Calculating a simple arithmetic answer with 3 pages of unnecessary manipulation, including extraordinarily long series of unnecessary expansion steps, greatly increasing the chances of a mistake, is a fiasco. It's not even math.
I think the power of math practice is working these out long hand, verbally calling out as you go, with pencil on paper. You develop so many other skills through this process. Agreed time consuming but worth the time especially if you are a slower learner.
Another solution
First we know that √ is the opposite of ^
So √ 64^2 = 64
So √49^2 = 49
So √15^4 = 15^2
Because 4 - 2 = 2
So we have 64 + 49 + 15^2
And you know that 15^2 = 225
So 64 + 49 + 225 = 338
338 / 2 = 169
Can you do without pencil and paper?
Don't you have to distribute the square root to 2 as well?
√(a²+b²+c²) ≠ a+b+c
Pure coincidence?
@@gagwong8964,
in spite of all we see that it still works, at least for her.
A nice radical math complication.
Was the first substitution necessary ?
No substitutions are _necessary_; they just make the algebra easier. I like that he did the two substitutions; it breaks the problem down into smaller steps, making it more accessible to viewers - especially those that aren't practiced enough at algebra to follow skipped steps.
Супер мега задача! Thank you so much for it! It's perfectly cool
Thank you very much!
SO SIMPLE. DO THIS AND DO IT IN YOUR HEAD. If you know the rules. Solved it my head in seconds. Here's how. The square root is basically x^1/2 so using distribution that kills 2 from each exponent leaving [(64+49+225)/2] = (338/2)= 162. Easy
An interesting coincidence insofar as generally the square root does not distribute over terms. The numbers were probably chosen precisely because this coincidence is a trap.
I've done it exactly the same.
338÷2 are 169 btw
Is this a solution or substantiation?
That would give me writers cramp.....and why would I ever need to do it....
Siento que la pudo resolver fácilmente desde un inicio sin tanto rollo.
The solution made the problem seem more complicated. I don’t need a calculator nor do I need to use this method to find the same exact answer.
Square root can be written as into the power (1/2)....
= [((64^2)+(49^2)+(15^4))^1/2]÷2
Multiply the power with every number inside...
= [(64^2)^1/2 + (49^2)^1/2 + (15^4)^1/2]÷2
= (64+49+15^2)÷2
= (64+49+225)÷2
= 338/2
= 169
Done
You forgot about √2 as the denominator.
Why not resolve the square root first? First simplification gives you (64+49+15^2)/2
Step seven you forgot the clouser ')' but it doesn't affect the final answer.thanks for the solution....😊
169. Took me 20 seconds on my calculator. Simple arithmetic following the order of operations. No need for fancy math on this one
169 = 13^2 = 8^3 - 7^3
I could do this, if necessary, as I was taught long multiplication in primary school !
Should've used a^2 + b^2 + c^2 form
Ow! My brain!
So...all this is nice and dandy...show this example by going to the grocery store and getting dinnner....
There's a directethod that's so much easier, what you're doing is unnecessarily complected
i ❤ Mathematics
169
Nice
Now I know why we invented the calculator…
The only mistake was canceling the root with the square. Should have been written with +/- (a+b).
Surprised he didn’t end it with 169 = 13^2
Or start with root root…
I'm pretty bad with math and i can't do all this process. If the goal is solving this operation without using calculator, based on my math memories, wouldn't it be easier just rewrite as: 64+49+15^2/2?
No. A simple example √(4 + 4) ≠ 2 + 2
For this it's just coincidental that the numbers work out the same.
Математика как Поэзия! Им ночью снятся Арктики и Звездные Галактики, ведь математик- это Фантазёр!
Are we not doing √ as the last step to get ±169 as answer?🤔
No, a square of number is positive
In algebra, the radical sign typically indicates the positive root only. A ± sign in front would indicate both roots.
(y = sqrt(x) has no negative values for y.)
However, I wouldn't say it's wrong if you include -169 as a second answer. I actually like that since the problem presents as arithmetic.
Thank you both for clearing my doubt.🙏🏻 I indeed went back to read about the basics . It says something like this : square root of a number is always positive, only in functions or equations like y=x², x has two values, the positive and the negative.
@@gagwong8964 I think that's basically correct. But y = x² seems like a strange example. A better example would be 9 = x². Here x = ±3. Or a = x². Here x = ±sqrt(a). Whereas, y=x² should be interpreted as a function in which there are infinite values for x.
@@go_gazelle Quadratic equations have two roots.
Depends on where you live.
Why not calculate directly.
Très astucieux. Bravo !
Thank you very much!
±169
This is way too long, I did it within 20 seconds, square root a number squared is just that number itself, square root of 64 squared is 64 and soo too is 49, square root of 15 to the power of 4 is 15 to the power of 2, square root is to the power of a half, so half the power of 4 to 2, it becomes 15 squared which is 225, 64 + 49 + 225 = 338, half of that is 169
If you change any of the numbers, then it doesn't work.
You can’t simply take the square root of each part of the expression and and add them to get the square of the answer in any but this specific case, where the third factor is the sum of the first two. As above, if you change any of the numbers, that method doesn’t work.
No, this is too long..sometimes question like this in a test only give you 1 to 2 minutes to solve,.
8
√64^2+49^2+15^4/2 ok
Then, you can kick to root 😁, then outstanding is 64+49+15^2/2, 64+49+225/2, 338/2 = 169 ✔️ simple 😂😂😂😂😂
Change any of the numbers and that "trick" doesn't work
Right Answer 169
This is why I changed my major to music!
College mathematics is just different. I loved exponentials as a kid did not learn of the different elevations until later. I know equal and above but not not learn much of below. But only learned to the end of highschool.
Brilliant!
Nice writing skills.😂
Клас!
Ans = 169 or even 13^2
Математика это жизнь!!!
Moral: when learning math, don't use TH-cam. 😂
Good old Pascal
If you took the square root and simplified the values under the radical sign, you will get:
(sqrt(64)^2) = 64
(sqrt(49)^2) = 49
(sqrt(15)^4) = 15^2 = 225
64 + 49 + 225 338
---------------------- = ------- = 169
2 2
The answer is 169
How to make 10 minutes video when you have content for 2.5 minutes !
Can we complicate things? Let's do it! Or... how to make things worse than they really are...
Cool. I loved it.
Wer brauch sowas
OMG
Легче все перемножить , будет корень из 28561=169.
So much simpler than using a calculator 😂
15×15
169. 5 minutes in mind.
169 can come in just 10-15sec. 😂😂😂😂😂😂😂😂
Guys, I just took the sqrt and squared exponents and cancel them out, then it'll be 64+49+(15)^2/2
Secondly, 64+49+225/2
Lastly, 338/2
Now you have your answer of 169, i didn't even learn this at school yet💀
The denominator wasn’t squared
I hope you pay attention in Maths class when your class gets up to this. 🤦🏻♂️
I didn't even learn this at school yet lol
🤦♂️Unexplainble, but you got the right answer
I did the same!
120√2 solution 😢
Lost me at 2.40 😮
I know
Very nice
So nice, Thank you very much!!!
140
I’m not advanced by any means but so many unnecessary steps. Neat little arithmetic problem though
Good Grief what a colossal waste of time. You could just multiply everything out by hand and take the square root. You would be finished in half the time. This is an example of a pointless "text book" problem. 4:47
WHY NOT ASK CHATGPT TO USE DIRECT COMPUTATION AND ALGEBRAIC METHOD. NONE AS DIFFICULT AS ABOVE METHOD.
15× rootover 226..?
Good lord this is not even close to being an efficient solution.
Difficult and lengthy
Fram ÍA
😂 0:31
Y no le veo ningún aporte a las matemáticas
Giochi di teoria , con calcolatrice vai subito alla soluzione😊 comunque ottima conoscenza👍
13²
328
Thanks! Step 6 you lose )
just calculated in my head and got the result of 169 without draw more than a full papersheet the root of 64 and 49 squared is just 64 and 49 plus 15 squared ist 225 338 half is 169 it was less than 1 minute
Same here wrote my comment then came to see how other people are screwing this up.
You didn't square root the denominator. It's just a coincidence that you have the same answer.
Use a calculator/phone
Calculator answer is 57.97 🧐
It's 15^4, not 15^2.
Prefiero seguir programando en asembler
You did too complicated.
There is easy way without Calculator
Too much steps
I expect he did that for instructional purposes, and to make the solution accessible to viewers with less practice in algebra.
not really, it's doable compared to sqrt thousands lol
Haaaaa bic mac
sqr30705
😂😂😂😂😂😂