radical

A radical is an expression that uses the root symbol √ to denote a root, such as sqrt(2), cbrt(x), or the fourth root of 81. The symbol itself is the radical sign; the whole expression built from it — sign, index, and the quantity inside — is the radical.

Three parts make up a radical. The radical sign √ tells you to take a root, the small index says which root (2 for square root, often left unwritten; 3 for cube root, and so on), and the radicand is the quantity tucked under the sign. So in the cube root of 7, the index is 3 and the radicand is 7.

Radicals let you write irrational quantities exactly rather than as rounded decimals. We simplify them by pulling perfect-power factors out from under the sign — for instance sqrt(12) = sqrt(4 × 3) = 2·sqrt(3) — and by rationalizing denominators. A radical is considered simplified when the radicand has no leftover perfect-power factors and no radical remains in a denominator.