# 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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