\magnify{3000}
\newbox\rootbox
\def\root#1\of{\setbox\rootbox=\hbox{$\scriptscriptstyle{#1}$}
  \mathpalette\finishroot}
\def\mathpalette#1#2{\mathchoice{#1\displaystyle{#2}}
  {#1\textstyle{#2}}{#1\scriptstyle{#2}}{#1\scriptscriptstyle{#2}}}
\def\finishroot#1#2{\setbox0=\hbox{$#1\sqrt{#2}$}
  \setdimen0=.6ht0 \advdimen0 by-.6dp0
  \mskip4mu \raise1dm0\copy\rootbox \mskip-9mu \box0}
\def\vphantom{\phant\yes\no}
\def\hphantom{\phant\no\yes}
\def\phantom{\phant\yes\yes}
\def\phant#1#2{\let\ifvph#1\let\ifhph#2
  \ifmmode\def\next{\mathpalette\mathphant}\else\let\next\makephant
  \fi\next}
\def\makephant#1{\setbox0=\hbox{#1}\finphant}
\def\mathphant#1#2{\setbox0=\hbox{$#1{#2}$}\finphant}
\def\finphant{\hbox{\ifvph\vrule height1ht0 depth1dp0 width0pt\fi%
    \ifhph\hskip1wd0\fi}}
\tracingall\tracingonline0
\hphantom{---}$\displaystyle\vphantom{\phantom{\sum}}\phantom\sum\showlists
$\displaystyle\root3\of2+\textstyle
\root3\of2+\scriptstyle\root\raise2ex\hbox{$\scriptstyle3$}
  \of{2\phantom{y}}+\scriptscriptstyle\root\overline3\of2$
$$\root\alpha+\beta\of{1+\root4\of{1+\root n\of{a\over b}}}$$
\bye