Root Calculator

About Roots

The n-th root of a number x is the value r such that r^n = x. The most familiar roots are the square root (n = 2) and cube root (n = 3). Roots are inverse operations of exponentiation — just as squaring gives the area of a square side, the square root recovers the side from the area.

For even-degree roots (n = 2, 4, 6, …), every positive number has two real roots: one positive (the principal root) and one negative. Negative numbers have no real even-degree roots — the result is a complex number involving i = √(−1). For odd-degree roots (n = 3, 5, 7, …), every real number (positive or negative) has exactly one real root of the same sign.

Roots can be expressed as fractional exponents: ⁿ√x = x^(1/n). This equivalence is powerful because it allows root operations to follow all the standard laws of exponents. For example, ⁴√(x³) = x^(3/4) and √x × ∛x = x^(1/2) × x^(1/3) = x^(5/6).