A very tricky Question from Cambridge University Admission Interview

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Muito facil

[@walterwen2975]

Cambridge University Admission Interview: (√2 + 1)ˣ + (√2 - 1)ˣ = 34; x =?
Let: a = √2 + 1, b = √2 - 1, a, b ϵ R⁺; a - b = 2, ab = 1, b = 1/a, a = 1/b
(a - b)² = a² + b² - 2ab = 2², a² + b² = 4 + 2ab = 4 + 2 = 6
(a² + b²) = a⁴ + b⁴ + 2a²b² = 6², a⁴ + b⁴ = 36 - 2ab = 36 - 2 = 34
(√2 + 1)ˣ + (√2 - 1)ˣ = 34 = a⁴ + b⁴ = 1/(b⁴) + 1/(a⁴) = a⁻⁴ + b⁻⁴; x = ± 4
Answer check:
(√2 + 1)⁴ + (√2 - 1)⁴ = a⁴ + b⁴ = a⁻⁴ + b⁻⁴ = 34; Confirmed as shown
Final answer:
x = 4 or x = - 4

[@KyyTyy]

t+(1/t)=34