I am looking at the following problem and I am struggling to follow the steps involved. Consider the non-interacting Hamiltonian $$H_{AB}=H_A\otimes I_B+I_A\otimes H_B$$
So I'm trying to prove that the unitary evolution of the joint state is given by $$|\psi\left(t\right)\rangle_{AB}=e^{-iH_At}\otimes e^{-iH_Bt}|\psi\left(t=0\right)\rangle_{AB}$$
Where $|\psi\rangle_{AB}=|\psi\rangle_A \otimes |\psi\rangle_B$
My working so far is
$$|\psi\left(t\right)\rangle_{AB}=e^{-i\left(H_A\otimes I_B+I_A\otimes H_B\right)t}|\psi\left(t=0\right)\rangle_{AB}$$
$$=e^{-i\left(H_A\otimes I_B\right)t}e^{-i\left(I_A\otimes H_B\right)t}|\psi\left(t=0\right)\rangle_{AB}$$
As the two Hamiltonians for the two system commute, so from here I am a little confused, I know the next step must be
$$=\left(e^{-iH_A t}\otimes I_B\right)\left(I_A\otimes e^{-iH_B t} \right)|\psi\left(t=0\right)\rangle_{AB}$$
But this isn't at all obvious to me why this is the case? I'm also not sure if this is off topic here and would be better suited to maths stack exchange so I apologise in advance.
Throughout I have set $\hbar=1$.
At this point I think I should probably use the definition of the matrix exponential as a Taylor series but I'm not sure.