Given a hyperreal infinitesimal number $\epsilon$ , is it meaningful to take its square root, $\sqrt{\epsilon}$ or any other root? What about using it as an exponent, as in $2^{\epsilon}$ ? And what about something like $\epsilon^{\epsilon}$ .
I've looked in a few sources like Henle's Infinitesimal Calculus but couldn't find anything on how or whether exponentiation works with hyperreal infinitesimals.
As mentioned in the comments, any real function extends canonically to a hyperreal function, sometimes called the natural extension. You can then apply Łoś's theorem/the transfer principle to apply any identities you know for the real functions to their hyperreal extension.
From the book you mentioned, Henle's Infinitesimal Calculus, Chapter 3:
Suppose that $f$ is a function on the reals in our language $L.$ How do we define $f$ on the hyperreals? Suppose that $j$ is a hyperreal, What is $f(j)$?
Well, $j$ is the sequence $j(1), j(2), j(3), j(4), \dots,$ so we define $f(j)$ to be the new sequence
$$f(j(1)), f(j(2)), f(j(3)), f(j(4)), \dots$$
Here is a possible problem: suppose that $j$ and $k$ represent the same hyperreal. Then $f(j)$ and $f(k)$ should represent the same hyperreal. Do they? Certainly, for let
$$A=\{n\mid j(n)=k(n)\}$$
and
$$B=\{n\mid f(j(n))=f(k(n))\}.$$
We know $A$ is quasi-big and we can easily check that
$$A\subseteq B.$$
Thus $B$ must be quasi-big so so $f(j)$ and $f(k)$ represent the same number. In this case $f$ was a 1-place function, but 2-place, 3-place, and other functions work in the same way.
