Indices that do not explode

Subscripts and superscripts are the first math syntax everyone learns in LaTeX, and also the source of the first confusing output everyone produces. The underscore _ lowers what follows, the caret ^ raises it, and both apply to exactly one character unless you say otherwise. That one-character rule is behind almost every mangled index you will ever see.

The basic forms

x_i x^2
x_{ij} x^{2n}
x_i^2 x_{i_j}^{2n}

x_i puts a single i in the subscript position and x^2 raises a single 2. Because _ and ^ grab only the next token, anything longer than one character has to be wrapped in braces: x_{ij} subscripts both letters, and x^{2n} raises the whole 2n. You can attach both a subscript and a superscript to the same symbol, in either order, and LaTeX stacks them correctly: x_i^2. Scripts also nest. In x_{i_j} the j is a subscript of the i, which is itself a subscript of x, and each level is set progressively smaller.

Multi-character scripts must use braces: x_ij is x_i then j. LaTeX does not report this as an error, because x_i followed by a variable j is perfectly legal math. The formula compiles, looks nearly right at a glance, and means something different. This silent near-miss is why the braces habit is worth building early: when in doubt, brace the script.

The same trap appears in limits and bounds. \sum_i=1^n does not do what it looks like; the correct form is \sum_{i=1}^{n}, where the entire i=1 is one braced subscript. If you use _ or ^ outside math mode entirely, you get the classic error covered in Missing dollar inserted.

Primes and dots

f' f'' \dot{x} \ddot{x}

For derivatives you rarely want a literal superscript. The apostrophe key produces a prime: f' renders as f with one prime, f'' with two, and LaTeX handles the raising and spacing itself, so there is no need for f^{\prime}. For time derivatives in the physics style, \dot{x} places a single dot over the x and \ddot{x} places two. These are accents, so they take their argument in braces.

One more subtlety: if you need a superscript after a prime, just chain them, as in f'^2. LaTeX combines them into a single superscript cluster.

Check your output

The fastest way to internalize the braces rule is to type x_ij and x_{ij} next to each other in the live playground and watch the difference render. Whenever an index looks subtly wrong in a compiled document, the first thing to check is a missing pair of braces.