  We study the heating time in periodically driven $D$-dimensional systems with
interactions that decay with the distance $r$ as a power-law $1/r^\alpha$.
Using linear response theory, we show that the heating time is exponentially
long as a function of the drive frequency for $\alpha>D$. For systems that may
not obey linear response theory, we use a more general Magnus-like expansion to
show the existence of quasi-conserved observables, which imply exponentially
long heating time, for $\alpha>2D$. We also generalize a number of recent
state-of-the-art Lieb-Robinson bounds for power-law systems from two-body
interactions to $k$-body interactions and thereby obtain a longer heating time
than previously established in the literature. Additionally, we conjecture that
the gap between the results from the linear response theory and the Magnus-like
expansion does not have physical implications, but is, rather, due to the lack
of tight Lieb-Robinson bounds for power-law interactions. We show that the gap
vanishes in the presence of a hypothetical, tight bound.
