Splitting root symbol

In cases such as the one you've brought up, I believe it's both simpler (for you) and clearer (for your readers) if you don't use the "surd" symbolism at all. Instead, use [...]^{1/2}, as is done in the following modified form of your MWE. Alternatively, as others have already suggested as well, you could move the long term into a "displayed" equation, i.e., onto a line by itself.

Not only is [...]^{1/2} simpler for you (because you don't have concern yourself with where the line break may fall), it is also easier on the readers' eyes (i.e., less distracting and hence also clearer) since the interline spacing in the paragraph in question doesn't have to be adjusted to accommodate the "surd".

\documentclass{standalone}
\begin{document}
\begin{minipage}{0.8\textwidth}
The \textbf{Euclidean distance} between two points $P=(x, y, z)$ and 
$Q=(a, b, c)$ in space is defined as $d(P, Q)= [(x - a)^{2} +(y - b)^{2} 
+ (z - c)^{2}]^{1/2}$. The distance between a point $P$ and a geometric 
object $S$, such as a line or plane, is the minimal distance $d(P, Q)$ 
which is possible with $Q\in S$.
\end{minipage}
\end{document}

This version allows TeX to break the math automatically. In this version you can only do this once a full version would need a counter and generate new unique names for the points each time.

\documentclass{article}

\makeatletter
\def\savepos#1{\leavevmode\pdfsavepos\write\@auxout{%
\gdef\string\save@#1{{\the\pdflastxpos sp }{\the\pdflastypos sp }}}}

\def\xx#1{\expandafter\expandafter\expandafter\@firstoftwo\csname save@#1\endcsname}
\def\yy#1{\expandafter\expandafter\expandafter\@secondoftwo\csname save@#1\endcsname}


\def\xsqrt#1{%
\sqrt{\vphantom{#1}}%
\ifx\save@L\@undefined
\else
\ifdim\yy{L}=\yy{R}%
\else
\rlap{$\overline{\vphantom{#1}\hskip\dimexpr\xx{b}-\xx{L}\relax}$}%
\fi
\fi
\savepos{L}#1\savepos{R}%
\ifx\save@L\@undefined
\else
\ifdim\yy{L}=\yy{R}%
\llap{$\overline{\hskip\dimexpr\xx{R}-\xx{L}\relax}$}%
\else
\llap{$\overline{\vphantom{#1}\hskip\dimexpr\xx{R}-\xx{a}\relax}$}%
\fi
\fi
}

\makeatother

\begin{document}



    \savepos{a}\begin{minipage}[t]{0.8\textwidth}
        The \textbf{Euclidean distance} between two points $P=(x, y, z)$ and $Q=(a, b, c)$
        in space is defined as $d(P, Q)= \xsqrt{(x - a)^{2} +(y - b)^{2} + (z - c)^{2}}$.
        The distance between a point $P$ and a geometric object $S$ like a line or plane is
        the minimal distance $d(P, Q)$ which is possible with $Q$ on $S$. 
    \end{minipage}\savepos{b}
\end{document}

\documentclass[preview,border=12pt]{standalone} % change it to your class!

\usepackage{mathtools}


\begin{document}
\begin{multline*}
A   = \sqrt{a+b+c} \\
    \overline{\rule{0pt}{2.5ex}+d+e+f}\\
    \overline{\rule{0pt}{2.5ex}+g+h+i}
\end{multline*}
\end{document}

Just another example:

\documentclass[preview,border=12pt]{standalone}% change it to your own document class!
\usepackage[a6paper,margin=1cm]{geometry}
\usepackage{mathtools}


\begin{document}
\begin{multline*}
d(x,y)  
    = \sqrt{(x-x_0)^2} \\
        \overline{\rule{0pt}{2.5ex}{}+(y-y_0)^2}\\
        \overline{\rule{0pt}{2.5ex}{}+(z-z_0)^2}
\end{multline*}
\end{document}