I'm currently doing some work with likelihoods. The following is a joint distribution $f_\theta (\mathbf{y})$ of $\mathbf{Y}$:
$$\begin{align} f_\theta (y_i) &= \prod_{i = 1}^n f_\theta (y_i) = \prod_{i = 1}^n \exp \left[ \sum_{j = 1}^k c_j(\theta) T_j(y_i) + d(\theta) + S(y_i)\right] \\ &= \exp \left[ \sum_{j = 1}^k c_j(\theta) \sum_{i = 1}^n T_j(y_i) + nd(\theta) + \sum_{i = 1}^n S(y_i) \right] \end{align}$$
I'm having a bit of difficulty understanding what was done with the products and sums to get from $\prod_{i = 1}^n \exp \left[ \sum_{j = 1}^k c_j(\theta) T_j(y_i) + d(\theta) + S(y_i)\right]$ to $\exp \left[ \sum_{j = 1}^k c_j(\theta) \sum_{i = 1}^n T_j(y_i) + nd(\theta) + \sum_{i = 1}^n S(y_i) \right]$. Would please take the time to explain this to me? Thank you.
$\prod\limits_{i=1}^{n} e^{a_i} =e^{ \sum\limits_{i=1}^{n} a_i}$. This is a property of the exponential function. Apply this and interchange the sums w.r.t. $i$ and $j$ (as mentioned in the comment by Mick below).