In Writing Math in LaTeX, we have already covered math mode, subscripts and superscripts, and the \frac command for building fractions. This fourth lesson out of seven means we are well past the halfway mark, and each new tool we pick up makes the previous ones more powerful. Today we turn to square roots and nth roots, learning how the \sqrt command wraps a radical sign around any expression — including fractions, exponents, and other nested elements you already know how to write.

The radical symbol is one of the most distinctive shapes in mathematical notation. Unlike a simple operator such as $+$ or $\times$, a root sign must stretch and scale to fully cover the expression inside it. In handwriting, we draw a horizontal bar that extends over the entire contents, but replicating this in plain text is nearly impossible.

LaTeX handles all of this automatically. No matter how wide or tall the expression under the radical becomes, the root symbol adjusts to fit. This is exactly the kind of detail that makes LaTeX ideal for technical writing.

The \sqrt command takes a single required argument in curly braces, and LaTeX wraps a radical sign around it.

For example, to write the square root of 2 in display mode:

This produces:

The expression inside the braces can be anything valid in math mode. Here are a few more examples:

Sometimes we need cube roots, fourth roots, or roots of any other index. The \sqrt command handles this with an optional argument in square brackets, placed before the curly braces.

As you may recall from the first course, optional arguments in LaTeX go inside square brackets [...], while required arguments go inside curly braces {...}. Here, the optional argument specifies the small index number that appears in the crook of the radical. For example:

This renders as:

One of the strengths of LaTeX is that commands compose naturally. We can place fractions, superscripts, subscripts, or any combination inside a \sqrt. For instance, to write the square root of a fraction:

This gives us:

In real formulas, roots appear alongside other terms. We can freely mix \sqrt with fractions, exponents, subscripts, and arithmetic operators. Consider the distance formula from coordinate geometry:

A few small errors come up frequently when working with roots:

• Forgetting curly braces: Writing \sqrt x + y instead of \sqrt{x + y}. Without braces, LaTeX only captures the first token after the command, so you get $\sqrt{x} + y$ instead of $\sqrt{x}$.

In this lesson, we learned how to typeset square roots with \sqrt{...} and nth roots with \sqrt[n]{...}. We saw how LaTeX automatically scales the radical to fit whatever is inside — whether that is a single variable, a fraction, or an expression filled with subscripts and superscripts. We also combined roots with other notation to build real formulas like the distance formula and the golden ratio.

Head to the practice exercises to try these ideas yourself. You will start by writing basic square and nth roots, then nest a fraction inside a radical, and finish by assembling a complete compound expression from scratch. In the next lesson, we will explore standard math functions like $\sin$, $\cos$, and $\log$, adding yet another essential piece to your LaTeX toolkit.