Nested roots with common baseline to minimize unnecessary white space

I am teaching from a precalc text that included the following in regards to nested roots:

I thought this looked pretty nice so I tried to reproduce it, but I ran into a few issues:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\noindent Really ugly:
$\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}.$
Not quite as ugly but still not ideal:
$\sqrt{\smash[b]{2\sqrt{\smash[b]{2\sqrt{\smash[b]{2\sqrt{\smash[b]{2}}}}}}}}.$
\end{document}

Is there a way to reproduce the textbook example? In mine, the second example looks much closer to that in the text but still off a good bit. This post inspired my use of smash but I imagine I may not be using it correctly. Any ideas?

Pull out the scalerel magic, and employ parameters \depthgrowth and \heightgrowth that define the growth in the \sqrt depth/height per nesting. The first result is with \depthgrowth at 0pt, \heightgrowth at 1pt. For the second/third examples, \depthgrowth at 1pt, \heightgrowth at 1.5pt

\documentclass{article}
\usepackage{mathtools,scalerel}
\def\depthgrowth{0pt}
\def\heightgrowth{1pt}
\newsavebox\zbox
\newcommand\zsqrt[1]{%
\ignoremathstyle
\savebox\zbox{$#1\rule{0pt}{.7\baselineskip}$}%
\stretchrel*{\sqrt{\phantom{#1}\kern0.5pt}}%
{\rule[-\dimexpr\dp\zbox+\depthgrowth]{0pt}{%
\dimexpr\ht\zbox+\dp\zbox+\depthgrowth+\heightgrowth}}%
\kern-\wd\zbox\textstyle#1%
}
\begin{document}
$\zsqrt{34\zsqrt{23\zsqrt{2\zsqrt{2}}}}$
\def\depthgrowth{1pt}
\def\heightgrowth{1.5pt}
$\zsqrt{34\zsqrt{23\zsqrt{2\zsqrt{2}}}}$
$\zsqrt{\frac{3}{4}\zsqrt{\frac{2}{3}\zsqrt{2\zsqrt{\frac{1}{2}}}}}$
\end{document}

Bottom line: I just got lucky here. I found that the height of the cross-bar changes drastically for small changes in \vs argument.

\documentclass{article}
\usepackage{amsmath}
\newcommand\vs[1]{\rule{0pt}{#1}}
\begin{document}
$\sqrt{\vs{10pt}2\smash{\sqrt{\vs{8.2pt}2\smash{\sqrt{\vs{8.1pt}2\smash{\sqrt{2}}}}}}}$
\end{document}

• This definitely looks nicer. This is very nitpicky, does it not look like the first and last root signs have the same baseline but the middle two are off a fair amount? The textbook example seems to have an entirely constant baseline which seems to be quite difficult to obtain. Maybe I can fool around with the new command you gave and see if changing a few things will help. – Daniel W. Farlow May 24 '17 at 2:17
• @DanielW.Farlow Good catch. As I said, I was lucky to get this far, but I will keep trying. – Steven B. Segletes May 24 '17 at 2:21
• @DanielW.Farlow Please see revision. – Steven B. Segletes May 24 '17 at 2:52
• Wow, awesome work! I'm going to try to really understand your revision, but it looks great. Thanks! – Daniel W. Farlow May 24 '17 at 2:56