LaTeX Subscript and Superscript: _, ^, Braces, and Common Errors

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Single character subscript
$x_1$, $x_2$, $x_n$

% Multiple character subscript (requires braces)
$x_{12}$, $x_{n+1}$, $x_{max}$

% Variables with subscripts
$a_i$, $b_j$, $c_{ij}$

% Greek letters with subscripts
$\alpha_1$, $\beta_{n}$, $\gamma_{i,j}$

\end{document}

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Single character superscript
$x^2$, $x^n$, $x^*$

% Multiple character superscript (requires braces)
$x^{10}$, $x^{2n}$, $x^{n+1}$

% Common exponents
$e^x$, $2^n$, $10^{-3}$

% Special notations
$x^{\prime}$, $x^{\dagger}$, $x^{\ast}$

\end{document}

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Both subscript and superscript
$x_1^2$, $a_n^m$, $x_i^{j+1}$

% Order doesn't matter
$x_1^2 = x^2_1$

% Complex combinations
$x_{n+1}^{2m}$, $a_{ij}^{kl}$

% With operators
$\sum_{i=1}^n$, $\int_0^{\infty}$

% Tensor notation
$T_{\mu\nu}^{\rho\sigma}$

\end{document}

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Nested superscripts
$x^{2^n}$, $e^{x^2}$, $2^{2^{2^2}}$

% Nested subscripts
$x_{i_j}$, $a_{n_{k+1}}$

% Mixed nesting
$x_i^{j^k}$, $a_{m_n}^{p^q}$

% Using additional braces for clarity
${(x^2)}^3 = x^6$

% Tower notation
$2^{2^{2^{\cdot^{\cdot^{\cdot}}}}}$

\end{document}

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Inline limits
$\lim_{x \to 0} f(x)$

% Display style limits
\[\lim_{x \to \infty} \frac{1}{x} = 0\]

% Summation
\[\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\]

% Product
\[\prod_{k=1}^{n} k = n!\]

% Integration
\[\int_0^1 x^2 \, dx = \frac{1}{3}\]

% Multiple limits
\[\lim_{\substack{x \to 0 \\ y \to 0}} f(x,y)\]

\end{document}

\documentclass{article}
\usepackage{amsmath}
\usepackage{mhchem} % Better chemistry support
\begin{document}

% Basic chemical formulas
$\mathrm{H_2O}$, $\mathrm{CO_2}$, $\mathrm{H_2SO_4}$

% Isotopes
$^{14}\mathrm{C}$, $^{235}\mathrm{U}$, $^2\mathrm{H}$

% Ions
$\mathrm{Na^+}$, $\mathrm{Cl^-}$, $\mathrm{SO_4^{2-}}$

% With mhchem package (recommended)
\ce{H2O}, \ce{CO2}, \ce{H2SO4}
\ce{^{14}C}, \ce{Na+}, \ce{SO4^{2-}}

% Chemical equations
\ce{2H2 + O2 -> 2H2O}

\end{document}

\documentclass{article}
\usepackage{amsmath}
\usepackage{tensor} % For tensor notation
\begin{document}

% Vector components
$\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$

% Derivatives
$\frac{d^2y}{dx^2}$, $\frac{\partial^2 f}{\partial x^2}$

% Tensors
$g_{\mu\nu}$, $R^{\alpha}_{\beta\gamma\delta}$

% Four-vectors
$x^\mu = (ct, x, y, z)$

% Christoffel symbols
$\Gamma^{\lambda}_{\mu\nu}$

% Units
$10^{-9}\,\mathrm{m}$, $3.0 \times 10^8\,\mathrm{m/s}$

\end{document}

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Single prime
$f'(x)$, $y'$

% Multiple primes
$f''(x)$, $f'''(x)$

% Alternative notation
$f^{\prime}(x)$, $f^{\prime\prime}(x)$

% With subscripts
$x'_1$, $x''_n$

% Prime on subscript
$x_{n'}$, $x_{n''}$

\end{document}

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Pre-subscripts and superscripts
${}_a^b X_c^d$

% Tensor notation with prescript package
\usepackage{tensor}
\tensor[^a_b]{X}{_c^d}

% Manual spacing adjustments
$x_{\!n}$ % negative thin space
$x_{\,n}$ % thin space
$x_{\:n}$ % medium space
$x_{\;n}$ % thick space

% Phantom subscripts for alignment
\begin{align}
x_1 &= a \\
x_{\phantom{1}2} &= b
\end{align}

\end{document}

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Accents with subscripts
$\hat{x}_i$, $\tilde{y}_n$, $\bar{z}_k$

% Accents with superscripts
$\hat{x}^2$, $\vec{v}^T$, $\dot{x}^n$

% Combined
$\hat{x}_i^2$, $\tilde{\phi}_{nm}^{kl}$

% Wide accents
$\widehat{xyz}_1^2$, $\widetilde{ABC}_{ij}$

\end{document}

\documentclass{article}
\usepackage{fixltx2e} % For \textsubscript
\begin{document}

% In text mode
H\textsubscript{2}O is water.
E = mc\textsuperscript{2} is Einstein's equation.

% Or use math mode
H$_2$O is water.
E = mc$^2$ is Einstein's equation.

% Ordinals
1\textsuperscript{st}, 2\textsuperscript{nd}, 3\textsuperscript{rd}

% Footnote markers
Text\textsuperscript{a}, Reference\textsuperscript{1}

\end{document}

% Tensors - superscripts before subscripts
$T^{\mu\nu}_{\rho\sigma}$  % Correct
$T_{\rho\sigma}^{\mu\nu}$  % Less standard

% Derivatives - use consistent notation
$\frac{\partial^2 f}{\partial x^2}$  % Standard
$f_{xx}$                              % Alternative

% Units - use upright text
$10^{-3}\,\mathrm{m}$     % Correct
$10^{-3}\,m$              % Wrong (italic m)