Showing Sufficiency of a Statistic [duplicate]

Question

Suppose that $X_1, X_2, \cdot \cdot \cdot \ , X_n$ is a random sample from a continuous distribution with pdf $f_X(x;\theta) = \theta x^{\theta-1}$, for $\ 0\leq x \leq 1$. Show that $W=\prod_{i=1}^n X_i$ is a sufficient statistic for $\theta.$ Is the maximum likelihood estimator for $\theta$ a function of $W$?

Here's what I did:

$$P(X_1=x_1,...,X_n=x_n|W=w)=\cfrac{P(X_1=x_1,...,X_n=x_n,\prod_{i=1}^nX_i=\prod_{i=1}^nx_i)}{P(\prod_{i=1}^nX_i=\prod_{i=1}^nx_i)} \stackrel{redundancy}= \cfrac{P(X_1=x_1,...,X_n=x_n)}{P(\prod_{i=1}^nX_i=\prod_{i=1}^nx_i)}=\cfrac{\theta^n(\prod_{i=1}^nx_i)^{\theta-1}}{P(\prod_{i=1}^nX_i=\prod_{i=1}^nx_i)}$$

Here I am stuck. I am having difficulty breaking up the probability in the denominator. Thanks for the help.