\bigl( \bigr) in inline mode and in align environment

Question

The following, I believe, is a strange behavior of \bigl(<math content>\bigr) in inline math mode and in the align environment. Consider the following MWE:

\documentclass[letterpaper]{article}
\usepackage{amsmath,amssymb,mathtools}
\usepackage{fourier}
\begin{document}
Consider having an exponential expression raised to another power; for instance, $\bigl(a^2\bigr)^4$. Using the definition of $a^n$,
    \begin{align*}
    \bigl(a^2\bigr)^4 &= \bigl(a^2\bigr)\bigl(a^2\bigr)\bigl(a^2\bigr)\bigl(a^2\bigr) && \text{Here the base is $\bigl(a^2\bigr)$}\\
    &=\underbrace{(a\cdot a)(a \cdot a)(a \cdot a)(a \cdot a)}_{\text{$2 \cdot 4$ or 8  factors}}\\
    &=a^8
    \end{align*}
\end{document}

The difference is obvious if you zoom in. For example:

Note the parenthesis are of different heights. I suppose its due to the definition of the commands \bigl and \bigr. Can any one shed some insight into the matter. I know several users here prefer using the above rather than \left to \right so am refraining from these commands. I would also like a fix to the size of the parenthesis.

EDIT

It seems that the exponent, "superscript", is the one to blame here for its behaviour in the modes \textstyle and \displaystyle.

Answer 1

The brackets are the same size, but the superscripts 2 and 4 are lower in inline math than in display style