How to distribute product sign across base and exponent terms

[Devon79]

I am calculating the likelihood of a hierarchical model and am having trouble distributing the product sign.

Here are the two expressions that I'm interested in:

Product from j=1 to j_k of: (t_jk)^(sum(Z_ijk) + a_k - 1)

and

Product from k=1 to K of: (b_k)^(j_k*a_k).

The tricky part for me is distributing the product sign across the base and the exponent terms simultaneously. Is it possible?

Devon

 

[g_edgar]

[tex]
\prod_{j=1}^{J_k} t_{jk}^{(\sum_{l=1}^L Z_{ljk}) + a_k-1} =
\left(\prod_{j=1}^{J_k} \prod_{l=1}^L t_{jk}^{Z_{ljk}}\right)
\left(\prod_{j=1}^{J_k} t_{jk}\right)^{a_k-1}
[/tex]

[tex]
\prod_{k=1}^K b_k^{j_k a_k}
\;[/tex] probably not much to do with this