$\sqrt{\text{Nested} \sqrt{\text{root} \sqrt{\text{problem}}}}$

Question

Problem

In the PSQ How to prove $\sqrt{6+2\sqrt{7+3\sqrt{8+\cdots}}}=4$?, I observe that the vertical separations above and below the middle square root sign are uneven.

\sqrt{6+2\sqrt{7+3\sqrt{8+\cdots}}}

Screenshot taken using Firefox 84.0.2 (64-bit) on Ubuntu 20.04

Is there a way to make the two vertical separations look even?

Effort

It seems fine in \displaystyle when it's rendered from editor preview.

$$\sqrt{6+2\sqrt{7+3\sqrt{8+\cdots}}}$$

Screenshot taken using same browser on same OS as above

Answer 1

Another option is to use \mathstrut, eg $\sqrt{ 6+2\sqrt{ 7+3\sqrt{\mathstrut 8+\cdots}}}$, which is produced from $\sqrt{ 6+2\sqrt{ 7+3\sqrt{\mathstrut 8+\cdots}}}$.

Answer 2

One-line solution $\sqrt{\vphantom{\bigcup} 6+2\sqrt{\vphantom{\bigcup} 7+3\sqrt{\vphantom{\bigcup} 8+\cdots}}}$ using \vphantom{\bigcup}:
$\sqrt{\vphantom{\bigcup} 6+2\sqrt{\vphantom{\bigcup} 7+3\sqrt{\vphantom{\bigcup} 8+\cdots}}}$

Answer 3

tl;dr– Using \mathstrut or/and \displaystyle might help.

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Comparisons:$$ \begin{array}{rcc} \textbf{Normal:} & \sqrt{6 + 2\sqrt{7 + 3\sqrt{8+\cdots}}} & \left(1\right) \\ \textbf{With}~\texttt{\mathstrut}\textbf{:} & \sqrt{6 + 2\sqrt{7 + 3\sqrt{\mathstrut 8 + \cdots}}} & \left(2\right) \\ \textbf{With}~\texttt{\displaystyle}\textbf{:} & {\displaystyle \sqrt{6 + 2 \sqrt{7 + 3\sqrt{8 + \cdots}}}} & \left(3\right) \\ \begin{array}{r}\textbf{With}~\texttt{\mathstrut} \\[-25px]\textbf{and}~\texttt{\displaystyle}\end{array}\textbf{:} & {\displaystyle \sqrt{6 + 2 \sqrt{7 + 3\sqrt{\mathstrut 8 + \cdots}}}} & \left(4\right) \end{array} _{.} $$

$\mathrm{\TeX} :$

1. Normal:
   \sqrt{6 + 2\sqrt{7 + 3\sqrt{8+\cdots}}}

2. With \mathstrut:
   \sqrt{6 + 2\sqrt{7 + 3\sqrt{\mathstrut 8 + \cdots}}}

3. With \displaystyle:
   \displaystyle \sqrt{6 + 2 \sqrt{7 + 3\sqrt{8 + \cdots}}}

4. With \mathstrut and \displaystyle:
   \displaystyle \sqrt{6 + 2 \sqrt{7 + 3\sqrt{\mathstrut 8 + \cdots}}}

Answer 4

tl;dr– This answer plays with spacing just for fun.

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@PM2Ring's answer suggested \mathstrut, then @GNUSupporter8964民主女神地下教會's answer tried constructing a similar mechanism through \vphantom{\bigcup}.

So, in that spirit and just for fun, trying \vphantom with differing heights, mostly just to see what they look like: $$ {\def\Spacing#1{\vphantom{\rule{0em}{#1em}}}} {\def\Example#1#2{ {#2{\sqrt{6 + 2 \sqrt{7 + 3\sqrt{\Spacing{#1} 8 + \cdots}}}}}}} {\def\Row#1{\\ \text{#1} & \Example{#1}{} & \Example{#1}{\displaystyle}}} \begin{array}{lcc} \begin{array}{c}\textbf{Height} \\[-25px] \left(\texttt{em}\right)\end{array} & \begin{array}{c}\textbf{Rendering, without} \\[-25px] \texttt{\displaystyle} \end{array} & \begin{array}{c}\textbf{Rendering, with} \\[-25px] \texttt{\displaystyle} \end{array} \Row{0} \Row{0.5} \Row{0.75} \Row{0.9} \Row{1} \Row{1.1} \Row{1.25} \Row{1.5} \Row{2} \end{array} _{.} $$

Each line is $$ \boxed{ \begin{array}{l} \color{blue}{\overbrace{\texttt{\displaystyle}}^{\text{with or without}}} ~\texttt{\{\sqrt\{6 + 2 \\sqrt\{7 + 3} \\[0.5em] \texttt{\\sqrt\{} \color{red}{\texttt{\\vphantom\{\\rule\{0em\}}} \color{red}{\underbrace{\texttt{\{}x~\texttt{em\}}}_{\begin{array}{c}\text{height} \\[-25px] \text{value}\end{array}} \texttt{\}}} ~\texttt{8 + \\cdots\}\}\}\}\}} \end{array} }_{,} $$ with the coloring for the $\color{blue}{\texttt{\\displaystyle}}$ and ${\color{red}{\textbf{height}}}$ parts.

Answer 5

Info too long-winded for a comment:

I dislike question titles which are all MathJax and no text. I tend to open questions by right-clicking on their titles, and this is not possible with all-MathJax titles.

Firefox browser remedy
(may work with other browsers)

1. Locate mouse just to the right of the query

2. right-click and choose inspect element from the pop-up menu

3. At this point, I have always had the inspection windows open up, with the (lower left pane) having the focus specifically on a compressed portion of the html code

4. Assuming you have the same experience, you can leave the mouse-focus as is, right-click the mouse, and then choose expand all

5. In the expanded area, one or two lines after the line that now has the mouse focus, you should see ...href="...".

6. Place the mouse focus inside the link, mouse-right-click, and then choose Copy Link Address.

7. Then close the Developer Tool Panes, manually go to a blank webpage, and then paste in the copied Link.