\documentclass{article}
\usepackage{amsmath}

\makeatletter

\newdimen\rootboxwd
\newbox\rootmaxwdbox

\def\r@@t#1#2{\setboxz@h{$\m@th#1\sqrtsign{#2}$}%
\dimen@\ht\z@\advance\dimen@-\dp\z@
\setbox\@ne\hbox{$\m@th#1\mskip\uproot@ mu$}%
\advance\dimen@ by1.667\wd\@ne
\mkern-\leftroot@ mu
\rootboxwd\wd\rootbox
\setbox\rootmaxwdbox\hbox{$\m@th#1\mskip10mu$}
\ifdim\rootboxwd<\wd\rootmaxwdbox
\mkern10mu\kern-\rootboxwd
\fi
\raise.6\dimen@\copy\rootbox
\mkern-10mu\mkern\leftroot@ mu\boxz@}

\makeatother

\begin{document}

$\boxed{\sqrt1}$

$\boxed{\sqrt[]1}$

$\boxed{\sqrt[.]1}$

$\boxed{\sqrt[3]1}$

$\boxed{\sqrt[\alpha]1}$

$\boxed{\sqrt[m]1}$

$\boxed{\sqrt[33]1}$

$\boxed{\sqrt[333]1}$

$\boxed{\sqrt[3333]1}$  

\end{document}