How to vertically align underbraces

Question

I am trying to balance underbraces in the equation below, how can I ensure that both underbraces at the same depth?

\begin{equation}
y = \underbrace{ \left [ x^2 + r^2 (a^2+\frac{a^{0.3}}{b}) \right]}_\text{growth rate} \times  \underbrace{ C_0}_\text{initial}
\end{equation} 

Answer 1

One way to solve this is to insert a "vertical phantom" -- an object with a certain height and depth but no width and therefore invisible, hence its name "phantom" -- in the second \underbrace expression.

In the case at hand, an immediately ready argument for such a \vphantom is the tallest "math molecule" from the first \underbrace expression; this turns out to be the term \left(\frac{a^{0.3}}{b}\right).

I would also like to suggest that you encase both \underbrace expressions in their entirety in curly braces; this'll improve the horizontal spacing around the \times symbol.

Addendum, Dec. 2019: The word "initial" is wider than the material immediately above it. This causes an unnecessarily wide gap to open up between the material in large square brackets and C_0. In the second row of the following screenshot, a \mathclap directive is used to set the width of "initial" to 0, leading to a better amount of horizontal spacing.

\documentclass{article}
\usepackage{mathtools} % for '\mathclap' macro
\begin{document}
\begin{align}
y 
&= {\underbrace{%
     \left[ x^2 + r^2 \left(a^2+\frac{a^{0.3}}{b}\right) \right] 
    }_{\text{growth rate}}} 
    \times  
    {\underbrace{%
     \vphantom{ \left(\frac{a^{0.3}}{b}\right) } 
     C_0}_{\text{initial}}}\\
&= {\underbrace{%
     \left[ x^2 + r^2 \left(a^2+\frac{a^{0.3}}{b}\right) \right] 
    }_{\text{growth rate}}} 
    \times  
    {\underbrace{%
     \vphantom{\left(\frac{a^{0.3}}{b}\right)} 
     C_0}_{\mathclap{\text{initial}}}} 
     \qquad \text{with \texttt{\string\mathclap}}\notag
\end{align} 
\end{document}

Answer 2

For just two braces the following is perhaps too much, but with three or more items it would become quite inconvenient to guess the largest item and to repeat \vphantom several times.

\documentclass{article}
\usepackage{amsmath,mathtools}

\newcommand{\ubrace}[2]{{\underbrace{#1}_{#2}}} % to avoid weird spacings

\newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}% for the second example


\ExplSyntaxOn

\NewDocumentCommand{\multiubrace}{mmm}
 {% #1 = items, #2 = descriptions, #3 = separators
  % #1, #2 and #3 are lists of braced groups
  % #1 and #2 should have the same number of items, #3 one less
  \egreg_multiubrace:nnn {#1} {#2} {#3}
 }

\cs_new_protected:Nn \egreg_multiubrace:nnn
 {
  \int_step_inline:nn { \tl_count:n {#1} }
   {
    % print the items with appropriate \vphantom
    \ubrace
     {% the item
      \vphantom{#1} \tl_item:nn {#1} {##1}
     }
     {% the description
      \tl_item:nn {#2} {##1}
     }
    % the separator
    \tl_item:nn {#3} {##1}
   }
 }

\ExplSyntaxOff

\begin{document}

The example in the question
\[
y = \multiubrace
     {
      { \left [ x^2 + r^2 (a^2+\frac{a^{0.3}}{b}) \right] } { C_{0} }
     }
     {
      { \text{growth rate} } { \mathclap{\text{initial}} }
     }
     { {\times} }
\]

And a more complicated one % https://tex.stackexchange.com/a/711734/4427
\[
\pder{\mathbf{M}}{t}=-
\multiubrace
  {
   { \gamma \mathbf{M} \times \mathbf{B}_{\textup{eff}} }
   { \frac{\alpha}{M} \mathbf{M} \times \pder{\mathbf{M}}{t} }
   { \theta_i^{S H} c_i^j \mathbf{M} \times \mathbf{M} \times \hat{y} }
  }
  { {\textup{precession}}{\textup{damping}}{\textup{SHE STT}} }
  { {+}{+} }
\]

\end{document}