Unification

Algebra Level 5

\[\large { e }^{ { i\pi }/{ 8 } }+{ e }^{ { 3i\pi }/{ 8 } }=\sqrt { \dfrac { \alpha +\sqrt { \beta } }{ \gamma } } (a+bi)\]

The equation above holds true for positive integers \(\alpha ,\beta ,\gamma ,a,\) and \(b\), with \(\alpha ,\beta ,\) and \(\gamma \) square-free.

Find \({ \alpha }^{ a }\times { \beta }^{ b }\times \gamma\).

Clarifications:

  • \(e\) denotes Euler's number, \(e \approx 2.71828\).

  • \(i=\sqrt{-1}\).

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