Move sign under exponent

I could understand a desire to avoid the hole, but your readers would have a hard time in interpreting the expression if the exponent is completely above \times.

I'd possibly use (3) or (4), but would prefer (5), removing the \times altogether, or better yet (6).

Notice the \! in the exponent to avoid the other hole.

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{equation}
\left(\frac{k+1}{k^\frac{k}{k+1}}\right)^{\!k+1}\times\frac{1}{(2k+2)(2k+1)}
\end{equation}

\begin{equation}
\left(\frac{k+1}{k^\frac{k}{k+1}}\right)^{\!k+1}
\mspace{-\medmuskip}
\times \frac{1}{(2k+2)(2k+1)}
\end{equation}

\begin{equation}
\left(\frac{k+1}{k^\frac{k}{k+1}}\right)^{\!k+1}
\mspace{-1.5\medmuskip}
\times \frac{1}{(2k+2)(2k+1)}
\end{equation}

\begin{equation}
\left(\frac{k+1}{k^\frac{k}{k+1}}\right)^{\!k+1}
\mspace{-2\medmuskip}
\times \frac{1}{(2k+2)(2k+1)}
\end{equation}

\begin{equation}
\left(\frac{k+1}{k^\frac{k}{k+1}}\right)^{\!k+1}
\frac{1}{(2k+2)(2k+1)}
\end{equation}

\begin{equation}
\frac{1}{(2k+2)(2k+1)}
\left(\frac{k+1}{k^\frac{k}{k+1}}\right)^{\!k+1}
\end{equation}

\end{document}

I personally would not do that but since you are asking...

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{mathtools}
\usepackage{amsmath}
\usepackage{amsthm}

\begin{document}

\begin{equation}
    \left(\frac{k+1}{k^\frac{k}{k+1}}\right)^{\mathrlap{k+1}} \times \frac{1}{(2k+2)(2k+1)}
\end{equation}

\end{document}

I'm not sure if this is what you mean, but you can do this with the \rlap command.

Here is the code:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{mathtools}
\usepackage{amsmath}
\usepackage{amsthm}

\begin{document}

\begin{equation}
    \left(\frac{k+1}{k^\frac{k}{k+1}}\right)^{\rlap{$\scriptstyle{k+1}$}} \times \frac{1}{(2k+2)(2k+1)}
\end{equation}

\end{document}