Dealing with tensor products in an exponent

I just found this post because I was confused by the same step. But I think I got it now with the help of @lionelbrits post and @Chris2807's comment. Just adding this for completeness and maybe helping someone else struggling with this:

$$ \begin{align} e^{(H_A \otimes I_B)} &= \sum_{n=0}^\infty\dfrac{(H_A \otimes I_B)^n}{n!} \\&= I_A \otimes I_B + H_A \otimes I_B + \dfrac{1}{2}(H_A \otimes I_B)^2 + ... \\&= I_A \otimes I_B + H_A \otimes I_B + \dfrac{1}{2}(H_A \otimes I_B)(H_A \otimes I_B) + ... \\&= I_A \otimes I_B + H_A \otimes I_B + \bigg(\dfrac{1}{2}(H_A)^2 \otimes (I_B)^2\bigg)+ ... \\&= (I_A + H_A + \frac{1}{2}(H_A)^2 + ... ) \otimes I_B \\&= e^{H_A}\otimes I_B \end{align} $$

where I also dropped the -i and t and used that $(I_B)^n = I_B$ with $n\in \mathbb{N}$.