Math Olympiad|Japanese|Algebra.

[@suryanarayanabadithamani7686]

👌👌👌👏👏👏

[@togelian]

In the start of the video it sounded like you meant that (a^5 + b^5)(a^6 + b^6) = a^11 + b^11, but I'm glad you cleared that up at the end

[@pedagogia8038]

Magnifique ,thank you

[@key_board_x]

(a + b)² = a² + 2ab + b² → given: a² + b² = 2
(a + b)² = 2 + 2ab → given: a + b = 1
1 = 2 + 2ab
2ab = - 1
ab = - 1/2
Resume:
a + b = 1 ← this is the sum S
ab = - 1/2 ← this is the product P
a & b are the solution of the equation:
x² - Sx + P = 0
x² - x - (1/2) = 0
Δ = (- 1)² - [4 * (- 1/2)] = 1 + 2 = 3
x = (1 ± √3)/2
a = (1 + √3)/2 and b = (1 - √3)/2 → or → b = (1 + √3)/2 and a = (1 - √3)/2
(1 + √3)² = 1 + 2√3 + 3
(1 + √3)² = 4 + 2√3
(1 + √3)² = 2.(2 + √3)
(1 + √3)³ = (1 + √3)².(1 + √3)
(1 + √3)³ = 2.(2 + √3).(1 + √3)
(1 + √3)³ = 2.(2 + 2√3 + √3 + 3)
(1 + √3)³ = 2.(5 + 3√3)
(1 + √3)⁴ = [(1 + √3)²]²
(1 + √3)⁴ = [2.(2 + √3)]²
(1 + √3)⁴ = 4.(2 + √3)²
(1 + √3)⁴ = 4.(4 + 4√3 + 3)
(1 + √3)⁴ = 4.(7 + 4√3)
(1 + √3)⁸ = [(1 + √3)⁴]²
(1 + √3)⁸ = [4.(7 + 4√3)]²
(1 + √3)⁸ = 16.(7 + 4√3)²
(1 + √3)⁸ = 16.(49 + 56√3 + 48)
(1 + √3)⁸ = 16.(97 + 56√3)
(1 + √3)¹¹ = (1 + √3)⁸. (1 + √3)³
(1 + √3)¹¹ = [16.(97 + 56√3)].[2.(5 + 3√3)]
(1 + √3)¹¹ = 32.(97 + 56√3).(5 + 3√3) → you can deduce that:
(1 - √3)¹¹ = 32.(97 - 56√3).(5 - 3√3)]
= (1 + √3)¹¹ + (1 - √3)¹¹
= [32.(97 + 56√3).(5 + 3√3)] + [32.(97 - 56√3).(5 - 3√3)]
= 32.[(97 + 56√3).(5 + 3√3) + (97 - 56√3).(5 - 3√3)]
= 32.[(97 * 5) + (97 * 3√3) + (5 * 56√3) + (56 * 9) + (97 * 5) - (97 * 3√3) - (5 * 56√3) + (56 * 9)]
= 32.[(97 * 5) + (56 * 9) + (97 * 5) + (56 * 9)]
= 32.[2.(97 * 5) + 2.(56 * 9)]
= 64.[(97 * 5) + (56 * 9)]
= 64.[485 + 504]
= 64 * 989
= [(1 + √3)¹¹ + (1 - √3)¹¹]/2¹¹
= (64 * 989)/2^(11)
= 2^(6) * 989/2^(11)
= 989/2^(5)
= 989/32
a¹¹ + b¹¹ = 989/32

[@JJONLINEMATHSCLASSchannel]

👍👍👍

[@mkhan-893]

using substitution and quadratic equation, you can find 2a^2 - 2a - 1 = 0, and then solve for a, and then substitute and solve for b, to get a = -0.366, and b = 1.366 or vice versa and then just take the power 11 to get the answer to be 30.9

[@JJONLINEMATHSCLASSchannel]

👍

[@AnikeadeJustina-z7m]

Am sure I will understand this

[@silasmusa5634]

Algebra the queen of mathematics 😂😂😂😂😂😂😂😂😂😂😂