$n^{th}$ roots and $e$. In Churchill's book on complex variables, the $n^{th}$ root of $e$ is defined to be $e^{1/n}$. A comment is made that in this respect $e$ is treated differently than the $n^{th}$ roots of other complex numbers (in the sense that there are typically n roots of the nth root of a number in complex analysis rather than just one as in the case of $e$).
I am curious why $e$ is treated so differently. Is there an obvious reason/motivation why?
Edit: The section from Churchill is,
As anticipated earlier, we define here the exponential function $e^z$ by writing
$$ e^z = e^xe^{iy}\ \ \ \ \ \ (z = x + iy)\ \ \ \ \ \ \ \ \ (1)$$
where Euler's formula
$$ e^{iy} = \cos y + i\sin y$$
is used and $y$ is to be taken in radians. We see from this definition that $e^z$ reduces to the usual exponential function in calculus when $y=0$; and, following the convention used in calculus, we often write $\exp z$ for $e^z$.
Note that since the positive $n$th root $\sqrt[n]{e}$ of $e$ is assigned to $e^x$ when $x = 1/n$ ($n = 2,3,\ldots$), expression (1) tells us that the complex exponential function $e^z$ is also $\sqrt[n]{e}$ when $z = 1/n$ ($n = 2,3,\ldots$). This is an exception to the convention that would ordinarily require us to interpret $e^{1/n}$ as the set of $n$th roots of $e$.
 A: The natural exponential function is defined by 
$$\exp(z) = \sum_{n=1}^\infty {z^n\over n!}.$$
This is an entire function. It is not hard to show it has all of the expected properties.  
To define $z^w$ you must define something like
$$z^w = \exp(z\log(w)).$$
Unfortunately, the exponential function is $2\pi i$-periodic.  Therefore it is not 1-1, so the business of defining a logarithm function becomes tricky. You must choose a domain to restrict the exponential function to so it is 1-1. And there the trouble begins. But where the trouble begins, complex analysis begins in all of its beauty and elegance.  
I quote one of my grad school professors, Sidney Graham, who said, 
"There are those who say that the study of complex variables is the study of the logarithm function."  
A: The point is that we want "the" exponential function to be single-valued.  If you want to write $\exp(1/n)$ as ${\rm e}^{1/n}$, that singles out one "$n$'th root of e".  There are still $n$ $n$'th roots of e, it's just that only one of them is written as ${\rm e}^{1/n}$.
