  We study the creation and propagation of exponential moments of solutions to
the spatially homogeneous $d$-dimensional Boltzmann equation. In particular,
when the collision kernel is of the form $|v-v_*|^\beta b(\cos(\theta))$ for
$\beta \in (0,2]$ with $\cos(\theta)= |v-v_*|^{-1}(v-v_*)\cdot \sigma$ and
$\sigma \in \mathbb{S}^{d-1}$, and assuming the classical cut-off condition $
b(\cos(\theta))$ integrable in $\mathbb{S}^{d-1}$, we prove that there exists
$a > 0$ such that moments with weight $\exp(a \min{t,1} |v|^\beta)$ are finite
for $t>0$, where $a$ only depends on the collision kernel and the initial mass
and energy. We propose a novel method of proof based on a single differential
inequality for the exponential moment with time-dependent coefficients.
