  In this paper, we derive formal general formulas for noncommutative
exponentiation and the exponential function, while also revisiting an
unrecognized, and yet powerful theorem. These tools are subsequently applied to
derive counterparts for the exponential identity $e^{A+B} = e^A e^B$ and the
binomial theorem $(A+B)^n = \sum \binom{n}{k} A^k B^{n-k}$ when the commutator
$[B, A]$ is either an arbitrary quadratic polynomial or a monomial in $A$ or
$B$. Analogous formulas are found when the commutator is bivariate.
Furthermore, we introduce a novel operator bridging between the normal and
antinormal forms.
