Question

Jika anda genius coba kerjakan soal ini dengan cara termudah. Berapa hasil dari
(√2 + 1)⁶ + (√2 - 1)⁶

Answer 1

Verified answer

Jawab:
= 198

Penjelasan :
Permisalan
a = √2 + 1, b = √2 - 1

Karena kesekawanan
b = 1/a, a = 1/b

a⁶ + b⁶
= (a²)³ + (b²)³

Permisalan lagi
c = a², d = b²

= c³ + d³
= (c + d)(c² + d² - cd)
= (c + d)((c+d)² - 2cd - cd)
= (c + d)((c+d)² - 3cd)

Substitute back
= (a²+b²)((a²+b²)² - 3a²b²)
= (a²+b²)((a²+b²)² - 3(ab)²)

Karena b = 1/a, dan a = 1/b
= (a²+(1/a)²)((a²+(1/a)²)² - 3(a(1/a))²)
= (a²+(1/a)²)((a²+(1/a)²)² - 3)
Masukkan lagi
= (a²+b²)((a²+b²)² - 3)

Lalu cari nilai a² + b²
a² + b² = (√2 + 1)² + (√2 - 1)²
a² + b² = 2+2√2+1+2-2√2+1
a² + b² = 2+1+2+1
a² + b² = 6

Maka
(a²+b²)((a²+b²)² - 3)
= 6(6² - 3)
= 6(36-3)
= 6(33)
= 198

Lah jadi (√2 + 1)⁶ + (√2 - 1)⁶
= (3 + 2√2)⁻³ + (3 - 2√2)⁻³ ?
(xcvi)