Unification

Algebra Level 5

eiπ/8+e3iπ/8=α+βγ(a+bi)\large { e }^{ { i\pi }/{ 8 } }+{ e }^{ { 3i\pi }/{ 8 } }=\sqrt { \dfrac { \alpha +\sqrt { \beta } }{ \gamma } } (a+bi)

The equation above holds true for positive integers α,β,γ,a,\alpha ,\beta ,\gamma ,a, and bb, with α,β,\alpha ,\beta , and γ\gamma square-free.

Find αa×βb×γ{ \alpha }^{ a }\times { \beta }^{ b }\times \gamma.

Clarifications:

  • ee denotes Euler's number, e2.71828e \approx 2.71828.

  • i=1i=\sqrt{-1}.

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