Cost minimization of expenditure function

[claeskens.matteo]

I want to find the right side of the expenditure function. I have made a latex document to visualize the math better. I know that for cost minimization you'll have to use the Langrange equation. But I'm stuck when calculating the FOC. The summation is no problem but the product symbol is a pain in the ***. Any help in the right direction would gladly be appreciated.