How can I take the following exponent, for some real-valued number a? $$a^{3+2j-9k+3i}$$ over the field of quaternions, or any field for that matter? On wikipedia we are given the following formula,
which seems relatively straight forward, but I do not know how to find a numerical expansion for any real-number a.
(I'm assuming that $a>0$).
What axioms are you starting from for exponentiation? One natural approach is to notice that in the reals, for $a>0$ $$a^x = e^{x\log a},$$ so if you have a definition for $e^x$ (for instance using the above power series) it is natural to define exponentiation of $a$ in any field using this formula as well. If $x = x_r + v$ is the decomposition of $x$ into its real and pure quaternionic parts, then by the formula you've listed above (and setting aside the question of checking that the power series of $e^x$ does indeed converge for all quaternions $x$, and does so to the formula you've listed):
$$a^x = e^{x_r\log a}\left(\cos [\|v\|\log a] + \frac{v}{\|v\|}\sin[\|v\|\log a]\right).$$