For this I recommend siunitx, that ensures uniformity in numbers and units.
\documentclass{article}
\usepackage{amsmath}
\usepackage{siunitx}
\sisetup{
exponent-product={{\cdot}}, % double brace for avoiding the space
per-mode=symbol,
}
\begin{document}
\SI{3e8}{\meter\per\second} % long form
\SI{3e8}{m/s} % abbreviated form
\SI[per-mode=reciprocal]{3e8}{\meter\per\second} % long form
\end{document}
Note that it's easy to change from a representation to another, when the long form is used (which is recommended).
Also this has the advantage that you just need to change the option in the preamble if you change your mind about how to represent that product.

The same effect can be obtained with the option tight-spacing, which however will act also on all binary operations, for instance uncertainties.
\documentclass{article}
\usepackage{amsmath}
\usepackage{siunitx}
\sisetup{
exponent-product=\cdot,
tight-spacing,
per-mode=symbol,
}
\begin{document}
\SI{3e8}{\meter\per\second} % long form
\SI{3e8}{m/s} % abbreviated form
\SI[per-mode=reciprocal]{3e8}{\meter\per\second} % long form
\end{document}
$3{\cdot} 10^8$ m/skills the\mathbinspace before and after\cdot. – Phelype Oleinik Apr 18 '19 at 14:31