# How does splitting up the products work?

I have been given the following problem and starting steps but I am stuck on the first step

A measure of goodness of fit to estimate the mean $$\mu$$ and standard deviation $$\sigma$$ of a normally distributed random variable is maximum likelihood estimation. For sample $$y = (y_1,... y_n)$$ drawn from the normally distributed random variables, $$Y_i \sim > N(\mu,\sigma)$$, the likelihood is given by
$$p(y|\mu,\sigma)=\prod_{i=1}^{n}\big(\frac{1}{2\pi\sigma^2}\big)^{\frac{1}{2}} exp \big( - \frac{1}{2\sigma^2}(\mu-y_i)^2 \big)$$
Using the fact that $$e^{-a}e^{-b} = e^{-a-b}$$ for arbitrary variables a and b, show the negative log likelihood has the following form:
$$L_(y|\mu,\sigma) = -\log p(y|\mu,\sigma) +\frac{1}{2\sigma^2}\sum_{i=1}^n(y_i-\mu)^2$$

$$\prod_{i=1}^n(\frac{1}{2\pi\sigma^2})^{\frac{1}{2}}\prod_{i=1}^n exp\big( - \frac{1}{2\sigma^2}(\mu-y_i)^2 \big)$$

I don't see why this can be done because

$$(a1 *a2) (b1* b2) \ne (a1* a2) b1 ( a1 *a2) b2$$

What am I missing?

$$\sum_{i=1}^n a+y_i = \sum_{i=1}^na+\sum_{i=1}^n y_i$$
$$\prod_{i=1}^n a y_i = \prod_{i=1}^na\prod_{i=1}^n y_i$$