You can use

• in text-mode (needs \textrm or similar in math-mode)
  • \textasciicircum or
  • \^{},
• in math-mode
  • \hat{} (only this produces a circumflex),
  • \widehat{}, or
  • \wedge (∧).

• in a verb-like manner

  • \string^,
  • \char`\^,

  • \verb!^!:

    \verb!d <- dist(fascores, method = "euclidean")^2!
    

Overview

Code

\documentclass{standalone}
\usepackage{upquote}% getting the right grave ` (and not ‘)!
\begin{document}
\begin{tabular}{lcc}
    Input                   &       Text       &                 Math                  \\ \hline
    \verb|\string^|         &     \string^     &              $\string^$               \\
    \verb|\char`\^|         &     \char`\^     &              $\char`\^$               \\
    \verb|\verb!^!|         &     \verb!^!     &              $\verb!^!$               \\ \hline
    \verb|\textasciicircum| & \textasciicircum &                  ---                  \\
    \verb|\^{}|             & \^{} (e.g. \^a)  &                  ---                  \\ \hline
    \verb|\hat{}|           &       ---        &       $\hat{}$ (e.g. $\hat a$)        \\
    \verb|\wedge|           &       ---        &      $\wedge$ (e.g. $a\wedge b$)      \\
    \verb|\widehat{}|       &       ---        & $\widehat{\ }$ (e.g. $\widehat{abc}$) \\
\end{tabular}
\end{document}

Output

You add a circumflex accent correctly by doing as you have done with i\hat. The lack of a dot above the "i" is the correct behaviour. Multiple citations here. If you would prefer that Word incorrectly shows a dot as well as the accent, then that would be a different question, though I doubt that such behaviour is possible.

It is probably much easier if you use the equation editor for it:

You can use an equation for this: activate the Insert tab of the ribbon and select Equation > Insert New Equation.

Hats and similar parts are found in the Accent dropdown.