If $a_1,\dotsc,a_k$ are algebraic numbers that are linearly independent over $\mathbb{Q}$, then $e^{a_1},\dotsc,e^{a_k}$ are algebraically independent over $\mathbb{Q}$. This is the Lindemann–Weierstrass theorem. So, if $a$'s are algebraic (as in the current version of the question), then the answer to your question is "yes".
Boris Bukh
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