Given,
$$ e^{\hat{A}+\hat{B}} = e^{\hat{B}}e^{\hat{A}} $$
I then consider the series expansion of both exponentials. This then leads to a particular order of operation derived from the order of addition in the exponential.
Am I correct in deducing that operators when inside the exponential function can no longer be moved in sequence of addition i.e.
$$ \hat{A}+\hat{B} \ne \hat{B}+\hat{A} $$
(this would then apply by extrapolation to all operators inside functions since all functions can be expanded in series form)