Help understanding exponential in Feynman-Kac proof

Question

I am working through the proof of the Feynman-Kac formula in Johnson and Lapidus' book The Feynman Integral and Feynman's Operational Calculus. In one of the steps of the proof they write that $$e^{-t(H_0 + V + ||V||_\infty)} = e^{-t||V||_\infty} e^{-t(H_0 + V)}$$ where $e^{-t(H_0 + V)}$ is the heat semigroup with generator $-(H_0 + V) = \frac 1 2 \Delta - V$. The $L^\infty$ norm $||V||_\infty$ is just a constant, but I am confused as to why we are allowed to pull it out of the exponent. The operator $H_0 + V$ is not a bounded operator so it doesn't seem like we can use the standard result that $e^Ae^B = e^{A+B}$ where $A$ and $B$ are bounded and commute. Any help would be much appreciated. Thank you.