index of a radical

The index of a radical is the small number written in the notch of the root sign, telling you which root to take. An index of 2 means a square root, 3 means a cube root, 4 means a fourth root, and so on. It answers the question “the root that does what when raised to a power?”

By long-standing convention the index 2 is left unwritten — a bare √ always means a square root — so you only see an index of 3 or more written explicitly. The index is exactly the n in “nth root,” and it matches the denominator of the fractional exponent form: the nth root of b is b^(1/n).

The index also decides the rules about signs. An even index (2, 4, 6, …) requires a nonnegative radicand for a real result and gives a nonnegative principal value; an odd index (3, 5, 7, …) accepts any real radicand and returns a single real root of the same sign. Mixing up the index is a common cause of sign errors.