Nice-looking p-th roots

The amsmath package provides commands to tweak the position of the p, this example is from the amsmath documentation:

\sqrt[\leftroot{-2}\uproot{2}\beta]{k}

In line with @DavidCarlisle's suggestion, I also recommend you load the amsmath package and experiment a bit with the \leftroot and \uproot options to the \sqrt macro. Since your root-symbol, p, has a longer lower-right spur and as well as a lower overall height than the \beta symbol does in the example code given in amsmath package's user guide, you may want to use a value of 3 rather than 2 to get a decently balanced formula:

\documentclass{article}
\usepackage{fouriernc,amsmath}
\begin{document}
$\sqrt[p]{a}$ vs.\ $\sqrt[\leftroot{-3}\uproot{3}p]{a}$

$\sqrt[p]{\tfrac{1}{n}}$ vs.\ $\sqrt[\leftroot{-3}\uproot{3}p]{\tfrac{1}{n}}$
\end{document} 

Of course, what looks "right" to you will depend importantly on your preferences. Just experiment around to find out what looks best to you.

Quick and dirty answer: use superscript p in the pth root symbol, i.e. \sqrt[^p]{x}. Not elegant, but mostly good enough. This is also fast and easy if you're using something like LyX. For example: