\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz}

\makeatletter

\newdimen\sqrtminh
\newdimen\sqrtmaxh
\sqrtminh=10.4pt
\sqrtmaxh=36.4pt
\newcount\rootpianyi

\def\r@@t#1#2{\setboxz@h{$\m@th#1\sqrtsign{#2}$}%
%
\dimen@\ht\z@\advance\dimen@\dp\z@
\ifdim\dimen@>\sqrtminh
\ifdim\dimen@<\sqrtmaxh
\rootpianyi=3
\fi\fi
%
\dimen@\ht\z@\advance\dimen@-\ht\rootbox
\setbox\@ne\hbox{$\m@th#1\mskip\uproot@ mu$}%
\advance\dimen@ by1.667\wd\@ne
\mkern-\leftroot@ mu\mkern5mu
\mkern\rootpianyi mu
\raise\dimen@\copy\rootbox
\mkern-\rootpianyi mu
\mkern-10mu\mkern\leftroot@ mu
\boxz@}

\makeatother

\begin{document}

\[\sqrt[3]x\ \sqrt[3]1\ \sqrt[3]{(1)}\ \sqrt[3]{(1)^2}\ \sqrt[3]{\left(1\right)^2}\ \sqrt[3]{\sum}\ \sqrt[3]{\frac12}\ \sqrt[3]{\frac{(1)}{(2)}}\ \sqrt[3]{\sum_{i=1}^{n}x}\]

\[
\foreach \i in {2,4,...,32} {\sqrt[3]{\rule{2pt}{\i pt}}\ }
\]

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

\end{document}
