How to cancel the root but not the expression inside it?

With some guessing for the correct value:

\documentclass{article}
\usepackage{cancel}

\begin{document}

\[
\cancel{\vphantom{\sqrt{(a+b+c+d+f+g+h+e)^{\cancel{2}}}}\hspace{1.5em}}
\hspace{-1.5em}\sqrt{(a+b+c+d+f+g+h+e)^{\cancel{2}}}
\]

\end{document}

Is this really necessary? By the way, the equality

is generally false. You need the absolute value and your careless cancellation might give wrong ideas to your students.

A possibly better syntax:

\documentclass{article}
\usepackage{cancel}

\newcommand{\cancelroot}[1]{%
  \sbox0{$\displaystyle\sqrt{\vphantom{#1}}$}%
  \cancel{\phantom{\usebox0}\hspace{0.5em}}\hspace{\dimexpr-\wd0-0.5em}%
  \sqrt{#1}%
}

\begin{document}

\[
\cancelroot{(a+b+c+d+f+g+h+e)^{\cancel{2}}}+
\cancelroot{\Bigl(\frac{a}{2}+b+c+d+f+g+h+e\Bigr)^{\cancel{2}}}
\]

\end{document}

Some ideas

\documentclass{article}
\usepackage{cancel}
\begin{document}
test
\[\cancel{\sqrt{\left(a+b+c+d+f+g+h+e\right)^{\cancel{2}}}}\]
versus
\[\left((a+b+c+d+f+g+h+e)^{\cancel{2}}\right)^{\cancel{\frac12}}\]
or
\[\sqrt[\cancel{2}]{\left(a+b+c+d+f+g+h+e\right)^{\cancel{2}}}\]
or
\[\rlap{$\cancel{\quad\vphantom{\sqrt{()^2}}}$}\sqrt{\left(a+b+c+d+f+g+h+e\right)^{\cancel{2}}}\]
or
\[\cancel{\sqrt{\vphantom{()^2}}}\overline{\left(a+b+c+d+f+g+h+e\right)^{\cancel{2}}}\]
or
\[\rlap{\line(1,1){18}}{\sqrt{\left(a+b+c+d+f+g+h+e\right)^{\cancel{2}}}}\]

\end{document}