Belgium | A nice Math Olympiad Simplification | You Should Know this trick

[@sakthidharan8028]

Fantastic explanation

[@superacademy247]

Glad it was helpful!

[@wes9627]

x = (√3-1)/2 is the positive root of x^2+x-1/2 = 0 or x^2 = 1/2-x
x^4 = x^2-x+1/4 = 1/2-x-x+1/4 = 3/4-2x; x^8 = 4x^2-3x+9/16 = 2-4x-3x+9/16 = 41/16-7x
x^10 = x^2*x^8 = (1/2-x)(41/16-7x) = 7x^2-(97/16)x+41/32 = 7/2-7x-(97/16)x+41/32 = -(209/16)x+153/32
Thus, x^10 = (209+153)/32-209√3/32 = 181/16-209√3/32

[@walterwen2975]

A nice Olympiad Math Simplification: [(√3 - 1)/2]¹⁰ = ?
Let: a = (√3 - 1)/2; a² = [(√3 - 1)/2]² = (4 - 2√3)/4 = (2 - √3)/2 = 1/2 - a
a³ = a(a²) = (a)(1/2 - a) = a/2 - a² = a/2 - (1/2 - a) = (3/2)a - 1/2
a⁵ = (a²)(a³) = (1/2 - a)[(3/2)a - 1/2] = (3/4)a - (3/2)a² - 1/4 + a/2
= (5/4)a - (3/2)(1/2 - a) - 1/4 = (5/4)a - 3/4 + (3/2)a - 1/4 = (11/4)a - 1
[(√3 - 1)/2]¹⁰ = a¹⁰ = (a⁵)² = [(11/4)a - 1]² = (11/4)²a² - 2(11/4)a + 1
= (121/16)(1/2 - a) - (11/2)a + 1 = (121/32 + 1) - [(121/16) + (11/2)]a
= 153/32 - (209/16)a = 153/32 - (209/16)[(√3 - 1)/2] = (362 - 209√3)/32
Answer check:
[(√3 - 1)/2]¹⁰ = [(1.732 - 1)/2]¹⁰ = 0.366¹⁰ = 4.316(10⁻⁵)
(362 - 209√3)/32 = (362 - 361.999)/32 = 0.0014/32 = 4.316(10⁻⁵); Confirmed
The calculation was achieved on a smartphone with a standard calculator app
Final answer:
[(√3 - 1)/2]¹⁰ = (362 - 209√3)/32

[@ichdu6710]

even more easier to calculate a² x a², square ithis result again and multiply that with a² again.by using the binomial equations.