# Digits of two irrational numbers, given their power with fixed number of digits

I have $a, b \in \mathbb{R} \setminus \mathbb{Q}$, I want to know the result of $a^b$, but I don't know exact $a, b$ because I write them in numeric form. My question is how many digits of $a, b$ have to I know to get $a^b$ with a fixed number of significant digits?

For example $a = \pi, b = e$, I want $\pi ^ e$ with 10 significant digits: how many digits at least of $\pi$ and $e$ I need?

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The partial derivatives of $u^v$, evaluated at $u=\pi$, $v=e$, will tell you how a small inaccuracy in the values of $e$ and $\pi$ will affect the accuracy of $\pi^e$. – Gerry Myerson Nov 11 '12 at 12:35
@GerryMyerson That's a very good idea! After finding the partial derivatives I've rewritten them in form $df = f(u,v)du, df = g(u,v)dv$, then put $df \le 10^{-10}$ and found $du$ and $dv$ from them. It gave me then minimum precision to set to $u$ and $v$ – Blex Nov 22 '12 at 21:09