Radical Expression
A radical expression contains a root symbol (square root, cube root, etc.) applied to a variable or polynomial expression.
Formula
\sqrt{x}, \; \sqrt[n]{x} = x^{1/n}
Definition
A radical expression contains a root symbol (square root, cube root, or beyond) applied to a number, variable, or polynomial, called the radicand; it can be rewritten with a rational exponent, $\sqrt[n]{\text{expression}} = (\text{expression})^{1/n}$, connecting radical operations directly to the rules of exponents. For even indices the radicand must be non-negative for a real result, and simplifying means factoring perfect squares (or nth powers) out of the radicand. Formally, a radical expression is an element of an algebraic extension generated by a root of a polynomial, over $\mathbb{Q}$, $\sqrt{2}$ generates the degree-$2$ extension $\mathbb{Q}(\sqrt{2}) = \{a + b\sqrt{2} : a, b \in \mathbb{Q}\}$, and more generally n-th roots generate cyclotomic or Kummer extensions; Galois theory answers exactly which algebraic numbers can be expressed using radicals, namely those whose minimal polynomial has a solvable Galois group.
Example
$\sqrt{x + 3}$ is only defined for $x \ge -3$, since the radicand must be non-negative. $\sqrt{48x^3} = \sqrt{16 \cdot 3 \cdot x^2 \cdot x} = 4x\sqrt{3x}$ (for $x \ge 0$), pulling the perfect square $16x^2$ out from under the radical. The splitting field of $x^5 - 2$ over $\mathbb{Q}$ contains fifth roots of $2$ and primitive $5$th roots of unity, with Galois group of order $20$, which is solvable, so $x^5 - 2 = 0$ is solvable by radicals.
Key Insight
The word "radical" comes from the Latin for root: a square root asks what number times itself gives the radicand. Radical expressions are the building blocks of solvability by radicals, and the connection between radical extensions and solvable Galois groups is the central result of Galois theory, explaining why generic polynomials of degree $5$ and above cannot be solved by radicals at all.