Power of an equation number

You can use \tag*.

\documentclass{article}
\usepackage{amsmath}

\begin{document}

We have a standard equation
\begin{equation}
|a+b| \le |a|+|b| \label{standard}
\end{equation}
but also its generalization to different exponents
\begin{equation}
\sqrt[\uproot{2}p]{|a|^p+|b|^p}\le |a|+|b| \tag*{(\ref{standard})\makebox[0pt][l]{$^p$}}
\end{equation}
for every $p\ge1$.

We can refer to equation~\eqref{standard} and to equation~\eqref{standard}$^p$.

\end{document}

If p is a fixed exponent, it can easily be done with the \newtagform command from mathtools:

\documentclass{article}
\usepackage{mathtools} %
\newtagform{power}({)$ ^p $}
\counterwithin{equation}{section}

\begin{document}

\section{Fermat’s equation}

\usetagform{power}

\begin{equation}
  x^n + y^n = z^n \label{pdf}
\end{equation}
For $n = 2$, equation \eqref{pdf} is known as the \emph{Pythagorean triples} problem.

\end{document}