Radical Expression
AlgebraA 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 square root (or other root) symbol applied to a number or variable. The expression under the root sign is called the radicand.
Example
$\sqrt{x + 3}$, $\sqrt{4x}$, and the cube root of $(2x - 1)$ are radical expressions. $\sqrt{x + 3}$ is only defined for $x \ge -3$ (radicand must be non-negative for square roots).
Key Insight
The word "radical" comes from the Latin for root. A square root asks: what number times itself gives the radicand?
Definition
A radical expression contains a radical symbol with an index $n$ (square root has index $2$) applied to an expression, written as $\sqrt[n]{\text{expression}} = (\text{expression})^{1/n}$. For even indices, the radicand must be non-negative for a real result. Simplifying radicals involves factoring out perfect squares (or nth powers) from the radicand.
Example
$\sqrt{48x^3} = \sqrt{16 \cdot 3 \cdot x^2 \cdot x} = 4x\sqrt{3x}$ (for $x \ge 0$). The perfect square $16x^2$ is pulled out from under the radical.
Key Insight
Radical expressions can be converted to rational exponent form: $\sqrt{x} = x^{1/2}$, cube root of $x = x^{1/3}$. This connects radical operations to the rules of exponents.
Definition
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 $\mathbb{Q}(\sqrt{2}) = \{a + b\sqrt{2} : a, b \in \mathbb{Q}\}$. More generally, n-th roots generate cyclotomic or Kummer extensions. The question of which algebraic numbers can be expressed using radicals is answered by Galois theory: an algebraic number is expressible by radicals iff its minimal polynomial has a solvable Galois group.
Example
The splitting field of $x^5 - 2$ over $\mathbb{Q}$ contains fifth roots of $2$ and primitive $5$th roots of unity. Its Galois group has order $20$, which is solvable, so $x^5 - 2 = 0$ is solvable by radicals.
Key Insight
Radical expressions are the building blocks of solvability by radicals. The connection between radical extensions and solvable Galois groups is the central result of Galois theory, explaining why degree $\ge 5$ generic polynomials cannot be solved by radicals.