\documentclass{article}
\usepackage{amsmath}
\begin{document}

\newcommand\testsq{\[
\sqrt[3]2+\sqrt[3]{\sum}+\sqrt[b-a]{\frac BA}+\sqrt[3]{\sum_{i=1}^n}
\]}

\testsq

\makeatletter
\def\plainroot@#1\of#2{\setbox\rootbox\hbox{%
$\m@th\scriptstyle{#1}$}%
\mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}}
{\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup}
\makeatother

\testsq

\end{document}