How do I prove all constants to some exponential power belong to little-o of some function
Try proving that $5^{2n} = o((\lg \lg n)^n)$.  Write out the proof in detail, and make sure you  understand every step.

Then, try replacing the $5$ in the base with an arbitrary constant $c$.  Does every step still go through?  Hint: you should be able to extend the argument to handle an arbitrary constant $c$, with very little change.

If you're still stuck, edit the question to show us what you've tried, including showing us your proof that $5^{2n} = o((\lg \lg n)^n)$ and your reasoning about which steps are still valid when you replace $5$ with an arbitrary constant $c$.