double sqrt (surd) symbol

Here is a version that uses stacks. It is not currently set up to obey the smaller math styles, however.

\documentclass[class=book]{book}
% Run with xelatex

%%% fonts
\usepackage{libertine}

\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{stackengine}
\stackMath
%%% mathfonts
\usepackage{unicode-math}
\setmathfont{texgyrepagella-math.otf}
\usepackage{microtype}


% "221A is the surd symbol in unicode. \surd is not working, it shows a p
\usepackage{mathtools}
\NewDocumentCommand\dsqrt{o m}{%
  \IfNoValueTF{#1}{%
    \,\stackengine{0pt}{\char"221A}{\,\char"221A#2}{O}{l}{F}{F}{L}%
  }%
  {
    \,\raisebox{3pt}{$\scriptscriptstyle#1$}\mkern-13mu\stackengine{0pt}{\char"221A}{\,\char"221A#2}{O}{l}{F}{F}{L}%
  }
}

\begin{document}

\begin{equation}\label{pre1}
\dsqrt{a} = \pm\sqrt{a}
\end{equation}

\begin{equation}\label{pre1}
2\dsqrt{a} = \pm2\sqrt{a}
\end{equation}

\begin{equation}\label{pre1}
\dsqrt[n]{a} = \pm\sqrt[n]{a}
\end{equation}

\begin{equation}\label{pre1}
2\dsqrt[n]{a} = \pm2\sqrt[n]{a}
\end{equation}

And here is another: inline  $\dsqrt{y} =\pm \sqrt{y}$.

And here is another: inline  $2\dsqrt{y} =\pm 2\sqrt{y}$.

And here is another: inline $\dsqrt[n]{y} =\pm \sqrt[n]{y}$.

And here is another: inline $2\dsqrt[n]{y} =\pm 2\sqrt[n]{y}$.

\end{document}

If one wanted the surd to match the argument size (as occurs with the \sqrt), then this definition should be substituted:

\NewDocumentCommand\dsqrt{o m}{%
  \IfNoValueTF{#1}{%
    \,\stackengine{0pt}{\sqrt{\vphantom{#2}}}{\,\sqrt{#2}}{O}{l}{F}{F}{L}%
  }%
  {
    \,\raisebox{3pt}{$\scriptscriptstyle#1$}\mkern-13mu\stackengine{0pt}{%
      \sqrt{\vphantom{#2}}}{\,\sqrt{#2}}{O}{l}{F}{F}{L}%
  }
}

Notice the difference with an x vs. a y argument.

Here's another attempt; unfortunately, it seems that the “surd” symbol is not available in Unicode.

\documentclass{book}
% Run with xelatex

%%% fonts
\usepackage{amsmath}
\usepackage{mathtools}
\usepackage{amsfonts}
\usepackage{amssymb}

%%% mathfonts
\usepackage{unicode-math}
\setmainfont{TeX Gyre Pagella}
\setmathfont{texgyrepagella-math.otf}
\usepackage{microtype}

% "221A is the surd symbol in unicode. \surd is not working, it shows a p
\NewDocumentCommand\dsqrt{o m}{%
  \mathop{%
    \IfNoValueTF{#1}
      {\doublesurd}
      {\mathrlap{^{\scriptscriptstyle#1}}\doublesurd}%
  }%
  {}#2
}

\makeatletter
\newcommand{\doublesurd}{\mathpalette\double@surd\relax}
\newcommand{\double@surd}[2]{%
  \ooalign{$\m@th#1\char"221A$\hidewidth\cr$\m@th#1\,\char"221A$\cr}%
}
\makeatother

\begin{document}

\begin{equation}
\dsqrt{a} = \pm\sqrt{a}
\end{equation}

\begin{equation}
2\dsqrt{a} = \pm2\sqrt{a}
\end{equation}

\begin{equation}
\dsqrt[n]{a} = \pm\sqrt[n]{a}
\end{equation}

\begin{equation}
2\dsqrt[n]{a} = \pm2\sqrt[n]{a}
\end{equation}

And here is another: inline  $\dsqrt{y} =\pm \sqrt{y}$.

And here is another: inline  $2\dsqrt{y} =\pm 2\sqrt{y}$.

And here is another: inline $\dsqrt[n]{y} =\pm \sqrt[n]{y}$.

And here is another: inline $2\dsqrt[n]{y} =\pm 2\sqrt[n]{y}$.

\end{document}