Let us have positive irrational numbers $a$ and $b$ represented by functions $f_a,f_b\colon\mathbb{N}\to\mathbb{N}$ respectively such that $f_a(0)=\left \lfloor{a}\right \rfloor$ and $f_a(i)$, $i>0$ is the ith digit of $a$, and similarly for $b$.

Is there some "standard" way to compute $f_{a^b}$ from $f_a$ and $f_b$? I would like to compute the some digits of $a^b$ and I'm convinced there are algorithms for this but I can't find any.