Here is the code that produces t

Here is the code that produces them:

\documentclass[12pt]{article}

\def\newsqrt{\mathpalette\DHLhksqrt} % it defines the new \sqrt in terms of the old one
\def\DHLhksqrt#1#2{\setbox0=\hbox{$#1\sqrt{#2\,}$}\dimen0=\ht0
\advance\dimen0-0.45\ht0
\setbox2=\hbox{\hspace{-.014cm}\vrule height\ht0 depth -\dimen0}%
{\box0\lower0.55pt\box2}}

\def\ee#1{\setbox0=\hbox{\Large $e$}\dimen0=3\ht0
\setbox2=\hbox{$^{#1}$}%
{\box0\lower-1.75pt \box2}}

\def\e#1{\setbox0=\hbox{\large $e$}\dimen0=3\ht0
\setbox2=\hbox{$^{#1}$}%
{\box0\lower-1pt \box2}} %\e{x}+\e{-x^2+2x+7}=f(x)

\begin{document}

Do you notice a difference between $\sqrt{ \frac{x^2-1}{x+1} }$ and $\newsqrt{ \frac{x^2-1}{x+1} } $\ ?

How about between $f(x)=e^{x^2},$ \ $f(x)=\e{x^2},$ and $f(x)=\ee{x^2}$ \ ?

\end{document}