Is there any notation for general $n$-th root $r$ such that $r^n=x$? As we know that the notation for the $n$-th principal root is $\sqrt[n]{x}$ or $x^{1/n}$. But the principal root is not always the only possible root, e.g. for even $n$ and positive $x$ the principal root is always positive but there is also another negative root. E.g. consider $r^2=4$, then $\sqrt 4 =+2$, but $r=-2$ is also a valid solution. Since $x$ is a function of $r$ for some given $n$, so let 
$$r^n=x=f(r).$$
We have $r=f^{-1} (x)$. Here $r \neq \sqrt[n]x$ because $\sqrt[n]x$ is the principal root not the general. So is there any notation like $\sqrt[n]{\phantom{aa}}$ for the general $n$-th root of the equation $r^n=x$?

For a non-negative real number $x$ there is always a unique choice of non-negative real $n$-th root, which is usually denoted by $\sqrt[n]{x}$. Furthermore, if $x$ is negative there is a unique choice of $n$-th root if $n$ is odd and none if $n$ is even.
In short, if $x \geq 0$ is real and $n$ is even, then the only real $n$-th roots of $x$ are $\pm \sqrt[n]{x}$ (and you can use this symbol), while if $x$ is real and $n$ is odd there is only one $n$-th root of $x$, denoted $\sqrt[n]{x}$.

You should understand, though, that taking roots usually involves a choice. In particular, every non-zero complex number has exactly $n$ $n$-th roots.
Now, you could keep the above choices for real numbers, but in general there is no canonical choice of root1. What we usually say instead is something like: "let $w$ be an $n$-th root of $z$". The nice thing, though, is that the other roots are then easily recoverable, because they are all of the form
$$
\zeta_n^i w \qquad \text{for } i \in \{0,\dotsc,n-1\}
$$
where $\zeta_n$ is a primitive $n$-th root of unity, i.e. a complex number such that $\zeta_n^n = 1$ and $\zeta_n^m \neq 1$ for every $0 < m < n$. Again, a choice is involved here, but you can always take
$$
\zeta_n = e^{2\pi i/n}
$$
TL;DR: If you wish to denote the generic $n$-th root of a complex number $z$ you may probably get away with the notation $z^{1/n}$. Just bear in mind that in general this is inherently ambiguous and you should treat this symbol more like a place-holder for an actual $n$-th root of $z$ than as a number.
[1] Technically, one could still define a unique choice of $n$-th root e.g. by taking the root with least argument (in $[0,2\pi)$). While this convention (or a similar one) may be used in analysis, I've never seen it in algebra or number theory.
