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 $pge1$.

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}