How do you actually calculate numbers like $2^{\pi}$ or $\sqrt[\leftroot{-2}\uproot{2}160]{2}$ My question is very simple, but I started to wonder how does one calculate numbers like $2^{\pi}$ or $\sqrt[\leftroot{-2}\uproot{2}160]{2}$?
For example I know that: 
$$2^{3/2} = \sqrt{2^3}=\sqrt{8}\approx2.83,$$
which is easy to calculate. But what about the cases I gave as an example? How would one go about and calculate those numbers using nothing else but a pencil and paper without calculator allowed? 
Would we use some kind of series here to approximate these numbers? 
It is also a bit unclear to me what $2^{\pi}$ means. For example to me, $2^3$ means in words: "Multiply number $2$ three times by itself", so multiplying $2$ by $\pi$ times by itself feels a bit weird when you're used to having integers in the exponent.
Thank you for any help and clarifications :)   

How I would think about $2^{\pi}$ is the following:
Consider an infinite sequence $(a_{0},a_{1},a_{2}\ldots)$ such that:
$$\sum_{n=0}^{\infty}a_{n}=\pi$$
Then
$$2^{\pi}=2^{\sum_{n=0}^{\infty}a_{n}}=2^{a_0+a_1+a_2+\cdots}=2^{a_{0}}2^{a_{1}}2^{a_{2}}\cdots$$
Now $2^{\pi}$ can be represented as a infinite product:
$$2^{\pi}=\prod_{n=0}^{\infty}2^{a_{n}}$$
There are infinite number of equations for $a_{n}$ but here's a link for a few, if you want to know them.
http://mathworld.wolfram.com/PiFormulas.html
