Multiplicative Identity for Exponential Function of Quadratic Matrix Form

Question

We know that it is true for the exponential scaler manipulation $$e^{\frac{a + b}{c}} = e^{\frac{a}{c}}e^{\frac{b}{c}}.$$ However, suppose $\textbf{A} \in \mathbb{R}^r$, $\textbf{B} \in \mathbb{R}^r$ and $\textbf{C} \in \mathbb{R}^{r \times r}$, then I am confused whether it is still true for $$e^{(\textbf{A} + \textbf{B})^T \textbf{C}^{-1}(\textbf{A} + \textbf{B})} = e^{\textbf{A}^T\textbf{C}^{-1}\textbf{A}} e^{\textbf{B}^T \textbf{C}^{-1}\textbf{B}}$$ If it is true, how to derive from LHS to RHS? If it is wrong, what is the correct answer?