Japanese || Math Olympiad || Can you solve this ? || Square Root Simplification😊

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Japanese || Math Olympiad || Can you solve this ? || Square Root Simplification😊
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16 сен 2024

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Комментарии : 17

@AllPracticalOfficial Месяц назад

Nice 🎉

@superacademy247 Месяц назад

Thanks 🤗🥳

@paulortega5317 Месяц назад

The values are related to the diophantine equations x^2 = 3*y^2 + 1 and x^2 = 3*y^2 - 2 It is a little easier to explain in terms of (1 + sqrt(3)) / 2 but this also works for (sqrt(3) - 1) / 2. For n = even number, let f(n) = (1 + sqrt(3))^n / 2^(n/2) = (x(n) + y(n)) / 2^(n/2); the values will satisfy x^2 = 3*y^2 + 1 For n = odd number, let f(n) = (1 + sqrt(3))^n / 2^((n+1)/2) = (x(n) + y(n)) / 2^((n+1)/2); the values will satisfy x^2 = 3*y^2 - 2 x(n+2) = 2*x(n) + 3*y(n) y(n+2) = 1*x(n) + 2*y(n) You can quickly get to f(10) f(2) = (2 + sqrt(3))/2 f(4) = (7 + 4*sqrt(3))/4 f(6) = (26 + 15*sqrt(3))/8 f(8) = (97 + 56*sqrt(3))/16 f(10) = (362 + 209*sqrt(3))/32 Note 362^2 = 3*209^2 + 1

@superacademy247 Месяц назад

Thanks for your input 👍😎💯

@RyanLewis-Johnson-wq6xs Месяц назад

Input (1/2 (sqrt(3) - 1))^10 = 181/16 - 209/32 sqrt(3) Result True Plot Plot Logarithmic form 10 log(1/2 (-1 + sqrt(3)), 1/2 (sqrt(3) - 1)) = log(1/2 (-1 + sqrt(3)), 181/16 - (209 sqrt(3))/32)

@MARTINWERDER Месяц назад

What is interesting about this formula is the decimal value after the exponentiation ^10 . The result is almost zero: 4.3160 e-5 or 0.00004316 . If you were to exponentiate even further, e.g. ^15 , the final result would be much smaller: 0.0000002835

@mrinaldas9614 Месяц назад

It is easier and more intuitive to go sqaring upto the power 8 and then multiplying with the squared value.

@遠傳五華 9 дней назад

@RyanLewis-Johnson-wq6xs Месяц назад

It’s in my head.

@soundsoflife9549 11 дней назад

What if you stuck it in a row 10x then multiplied pairs so you had 5 pairs, then no srt3 of div 2 ect..

@SidneiMV Месяц назад

[(√3 - 1)/2]¹⁰ [(√3 - 1)/2]² = (4 - 2√3)/4 = (2 - √3)/2 a = √3 - 1 (a/2)² = (1 - a)/2 a² = 2(1 - a) a⁴ = 4(a² - 2a + 1) a⁴ = 4(2(1 - a) - 2a + 1) a⁴ = 4(3 - 4a) a⁸ = 16(16a² - 24a + 9) a⁸ = 16(32(1 - a) - 24a + 9) a⁸ = 16(41 - 56a) a¹⁰ = a⁸a² = 16(41 - 56a)2(1 - a) a¹⁰ = 32(56a² - 97a + 41) a¹⁰ = 32(112(1 - a) - 97a + 41) a¹⁰ = 32(153 - 209a) a¹⁰ = 32(153 - 209(√3 - 1)) a¹⁰ = 32(362 - 209√3) [(√3 - 1)/2]¹⁰ = (362 - 209√3)/32

@superacademy247 Месяц назад

Great 👌 method. Thanks 😊

@SidneiMV Месяц назад

@@superacademy247 I think it's very similar to yours ☺️

@RyanLewis-Johnson-wq6xs Месяц назад

Input (1/2 (sqrt(3) - 1))^10 = 181/16 - 209/32 sqrt(3)=11.3125-6.53125Sqrt[3] Result True Plot Plot Logarithmic form 10 log(1/2 (-1 + sqrt(3)), 1/2 (sqrt(3) - 1)) = log(1/2 (-1 + sqrt(3)), 181/16 - (209 sqrt(3))/32)

@key_board_x Месяц назад

(√3 - 1)² = 3 - 2√3 + 1 (√3 - 1)² = 4 - 2√3 (√3 - 1)² = 2 * (2 - √3) (√3 - 1)⁴ = [(√3 - 1)²]² (√3 - 1)⁴ = [2 * (2 - √3)]² (√3 - 1)⁴ = 4 * (4 - 4√3 + 3) (√3 - 1)⁴ = 4 * (7 - 4√3) (√3 - 1)⁸ = [(√3 - 1)⁴]² (√3 - 1)⁸ = [4 * (7 - 4√3)]² (√3 - 1)⁸ = 16 * (49 - 56√3 + 48) (√3 - 1)⁸ = 16 * (97 - 56√3) (√3 - 1)¹º = (√3 - 1)⁸ * (√3 - 1)² (√3 - 1)¹º = 16 * (97 - 56√3) * 2 * (2 - √3) (√3 - 1)¹º = 32 * (97 - 56√3) * (2 - √3) (√3 - 1)¹º = 32 * (194 - 97√3 - 112√3 + 168) (√3 - 1)¹º = 32 * (362 - 209√3) [(√3 - 1)/2]¹º = (√3 - 1)¹º/2¹º [(√3 - 1)/2]¹º = 32 * (362 - 209√3) / 2¹º [(√3 - 1)/2]¹º = 2⁵ * (362 - 209√3) / 2¹º [(√3 - 1)/2]¹º = (362 - 209√3) / 2⁵ [(√3 - 1)/2]¹º = (362 - 209√3) / 32

@superacademy247 Месяц назад

Thanks 👍💯 for your approach