I'm pretty rusty at on-paper arithmetic, but I managed to just math out the answer directly in 2 minutes and 18 seconds, compared to the video's 9:30 doing it "simplified": 11 seconds to hand-multiply 81^2 to get 6561 15 seconds to hand-multiply 25^2 to get 625 44 seconds to hand-multiply 14^4 to get 38416 16 seconds to hand-add 6561+625+38416 and divide by 2 to get 22801 5 seconds to see that 22801 is close to 225*100, so the square root is close to 15*10 (since 15^2 = 225) = 150. Using Newton's method for refining square root guesses (guess(N+1) = 0.5*(guess(N) + A/guess(N)) where A is the number you're trying to square root): 24 seconds to hand-divide 22801 by 150 to get ~152 3 seconds to calculate 0.5*(150+152) to get 151 20 seconds to square 151 (getting 22801) to ensure it's an exact answer 138 total seconds, or 2 minutes and 18 seconds.
How to complicate a mental arithmatic problem. The square root nullifies the 2 squared numbers(81 & 25) and reduces 14 to a simple square. Add them up, divide by two and you get 151. So simple and easy.
Except that the way you got to 151 is wrong so it's entirely coincidental. The radical doesn't distribute like that. Try doing the same thing you did, but replace 81²+25²+14⁴ with 3²+4²+5². Using your method: √{(3²+4²+5²)/2} = (√3²+√4²+√5²)/2 = (3+4+5)/2 = 12/2 = 6 You would get an answer of 6. But that is wrong. The correct answer is 5. Here's how the arithmetic plays out: √{(3²+4²+5²)/2} = √{(9+16+25)/2} = √(50/2) = √25 = 5
*=read as squre root ^=read as to the power 81^2=(9^2)^2=9^4 Let a=9 Simillarly 25^4=(5^2)^2=5^4 Let b=5 So 14^4=(a+b)^4 According to the formula (a+b)^4=a^4+b^4+6a^2b^2+4a^3b +4ab^3 The numerator of the question can be written as a^4+b^4+(a+b)^4 a^4+b^4+a^4+b^4+6a^2b^2+4a^3b+ 4ab^3 2a^4+2b^4+6a^2b^2+4a^3b+4ab^3 2(a^4 +b^4+3a^2b^2+2a^3b+2ab^3) 2(a^4+b^4+2a^2b^2+a^2b^2+2a^3b+ +2ab^3){2(a^4+b^4+2a^2b^2)}+{(ab)^2+2ab(a^2+b^2)} 2{(a^2+b^2)^2+(ab)^2+2ab(a^2+b^2)} 2{a^2+b^2+ab}^2 As per question the denominator is2 So the expression will be *{a^2+b^2+ab}^2 a^2+b^2+ab Now put the value of a, b 9^2+5^2+(9×5) 81+25+45 151(Ans) Due to some problem I have skipped few steps....
I don't what the heck I just watched nor why it was recommended to me but this was brilliant!! I guess youtube heard when I said I wanted more academic stuff 😂 I was trying to find calculations in biomechanics but this helps haha
Well, you would, if your algebra is to a sufficient standard for the exam you are doing (!) and provided you can spot the 2 clever dodges. He does however spend an extraordinary amount of time explaining some of the trivial stuff. And unfortunately I only spotted one of the two dodges, so would probably have struggled to get there !
@@KAF128 See my reply to the video, I did the straight math by hand to get the answer in about two minutes, while the video method took over nine minutes.
No, as per the rules of polynomials , a linear function can only have one value and not two , so (a+b)²^{1/2} = |a+b| U could treat it as a rule too That √x² = |x| for all x Both the domain and range of a square root function is [0,∞)
@@ExquisiteHappiness to expand a bit my knowledge in the universe of mathematics, can you explain to me what is "domain" and "range" of a function or a very specific kind of number?
@@elchile336 Now in order to get that, u should know what a function is It's like a combination of numbers, variables and many other like f(x) = x² - sinx + t - x Here in f(x), the x in the braket shows a particular value for the function 'f' and t is just a constant, now u can put any value of 'x' and and u would get an output denoted by f(x) Like, if x = 3 f(3) = 3² - sin3 + t - 3 So, for a value of x, u got a value of f(x) That is the domain and range , not clear? The input 'x' is the domain and 'f(x)' is the range All the values u can put as long as the function is defined, it's the domain of the function and range is all the possible values u could get For square function Let's say f(x) = √x In the real world, we can only put x from 0 to ∞, because negative numbers inside square roots aren't defined and the value of f(x) is also from 0 to ∞, it is never negative( many people make this mistake) Soo I hope it's clear to yaa~
I'm pretty rusty at on-paper arithmetic, but I managed to just math out the answer directly in 2 minutes and 18 seconds, compared to the video's 9:30 doing it "simplified":
11 seconds to hand-multiply 81^2 to get 6561
15 seconds to hand-multiply 25^2 to get 625
44 seconds to hand-multiply 14^4 to get 38416
16 seconds to hand-add 6561+625+38416 and divide by 2 to get 22801
5 seconds to see that 22801 is close to 225*100, so the square root is close to 15*10 (since 15^2 = 225) = 150.
Using Newton's method for refining square root guesses (guess(N+1) = 0.5*(guess(N) + A/guess(N)) where A is the number you're trying to square root):
24 seconds to hand-divide 22801 by 150 to get ~152
3 seconds to calculate 0.5*(150+152) to get 151
20 seconds to square 151 (getting 22801) to ensure it's an exact answer
138 total seconds, or 2 minutes and 18 seconds.
How to complicate a mental arithmatic problem. The square root nullifies the 2 squared numbers(81 & 25) and reduces 14 to a simple square. Add them up, divide by two and you get 151. So simple and easy.
Except that the way you got to 151 is wrong so it's entirely coincidental. The radical doesn't distribute like that. Try doing the same thing you did, but replace 81²+25²+14⁴ with 3²+4²+5².
Using your method:
√{(3²+4²+5²)/2} =
(√3²+√4²+√5²)/2 =
(3+4+5)/2 =
12/2 =
6
You would get an answer of 6. But that is wrong. The correct answer is 5. Here's how the arithmetic plays out:
√{(3²+4²+5²)/2} =
√{(9+16+25)/2} =
√(50/2) =
√25 =
5
@@Eternitycomplex It doesn't even give the right answer.
Well, except that (9 + 5 + 14^2)/2 = 105, not 151.
A fabulous video
Thank you very much ❤
Excellent solution
*=read as squre root
^=read as to the power
81^2=(9^2)^2=9^4
Let a=9
Simillarly
25^4=(5^2)^2=5^4
Let b=5
So 14^4=(a+b)^4
According to the formula
(a+b)^4=a^4+b^4+6a^2b^2+4a^3b
+4ab^3
The numerator of the question can be written as
a^4+b^4+(a+b)^4
a^4+b^4+a^4+b^4+6a^2b^2+4a^3b+
4ab^3
2a^4+2b^4+6a^2b^2+4a^3b+4ab^3
2(a^4 +b^4+3a^2b^2+2a^3b+2ab^3)
2(a^4+b^4+2a^2b^2+a^2b^2+2a^3b+
+2ab^3){2(a^4+b^4+2a^2b^2)}+{(ab)^2+2ab(a^2+b^2)}
2{(a^2+b^2)^2+(ab)^2+2ab(a^2+b^2)}
2{a^2+b^2+ab}^2
As per question the denominator is2
So the expression will be
*{a^2+b^2+ab}^2
a^2+b^2+ab
Now put the value of a, b
9^2+5^2+(9×5)
81+25+45
151(Ans)
Due to some problem I have skipped few steps....
Great and beautiful job 👍
الرياضيات هي واحد من العلوم التي يتمتع بها الانسان مهما اختلفت اللغة، اللون، الجنس، الدم، العرق...
Astonishing solution!
I don't what the heck I just watched nor why it was recommended to me but this was brilliant!! I guess youtube heard when I said I wanted more academic stuff 😂
I was trying to find calculations in biomechanics but this helps haha
Wow!
மிகவும் நன்று very nice
Шикарно! Дякую! Клас!
Очень интересное решение, спасибо
👍👏
❤❤❤❤❤.
How do you arbitrarily raise 9^2 to (9^2)^2? Without changing the problem ? 🤔
original problem had 81^2. 81 = 9^2 so 81^2 = (9^2)^2 = 9^4
No mistakes were made.
A nice solution, unfortunately in a test situation you wouldn’t have time to use it.
Well, you would, if your algebra is to a sufficient standard for the exam you are doing (!) and provided you can spot the 2 clever dodges. He does however spend an extraordinary amount of time explaining some of the trivial stuff. And unfortunately I only spotted one of the two dodges, so would probably have struggled to get there !
@@KAF128 See my reply to the video, I did the straight math by hand to get the answer in about two minutes, while the video method took over nine minutes.
When you divided top and bottom by 2, how did 3x^2y^2 become 2x^2y^2 in the next line?
He writes 3x²y² as 2x²y²+x²y² in the next line
Thats the pen that i use lol
なかなか凄い。私は日本人だが、やはり海外の数学の方が面白い。
🎉 excellent
What is a real life situation that this is useful knowledge?
None whatsoever, but it's good brain exercise.
At 8:20 can't we also consider that it can be = - a- b? Is it a mistake?
No, as per the rules of polynomials , a linear function can only have one value and not two , so (a+b)²^{1/2} = |a+b|
U could treat it as a rule too
That √x² = |x| for all x
Both the domain and range of a square root function is [0,∞)
@@ExquisiteHappiness to expand a bit my knowledge in the universe of mathematics, can you explain to me what is "domain" and "range" of a function or a very specific kind of number?
@@elchile336 Now in order to get that, u should know what a function is
It's like a combination of numbers, variables and many other
like f(x) = x² - sinx + t - x
Here in f(x), the x in the braket shows a particular value for the function 'f' and t is just a constant, now u can put any value of 'x' and and u would get an output denoted by f(x)
Like, if x = 3
f(3) = 3² - sin3 + t - 3
So, for a value of x, u got a value of f(x)
That is the domain and range , not clear?
The input 'x' is the domain and 'f(x)' is the range
All the values u can put as long as the function is defined, it's the domain of the function and range is all the possible values u could get
For square function
Let's say
f(x) = √x
In the real world, we can only put x from 0 to ∞, because negative numbers inside square roots aren't defined
and the value of f(x) is also from 0 to ∞, it is never negative( many people make this mistake)
Soo I hope it's clear to yaa~