\documentclass{article}

\begin{document}
An equation with different subscripts
\begin{displaymath}
a_r=a_\beta+a_{i+j}
\end{displaymath}
and the same with superscripts
\begin{displaymath}
a^r=a^\beta+a^{i+j}
\end{displaymath}
Now an equation that is in both
\begin{displaymath}
E=mc^3\ \ \ \ x=1+\sin y^{i+j+\cdots}
\end{displaymath}
and equation using \verb|\left| and \verb|\right|

\begin{displaymath}
S=\left    \{ 5+\smile+\frac{numerator}{denominator} \right \}
\end{displaymath}

\end{document}
