Root Calculator

What an nth Root Means

The nth root of a number x is the value r that satisfies rⁿ = x, written as ⁿ√x (or x^1/n). The most familiar cases are the square root (n = 2) and cube root (n = 3), but the same idea extends to any positive integer degree. For a positive radicand, this calculator computes the principal (positive) real root using r = x^1/n, evaluated as exp((1/n)·ln(x)) for accuracy. When the radicand is negative and n is odd, a real negative root exists (for example, the cube root of -8 is -2); when the radicand is negative and n is even, there is no real root, since no real number raised to an even power produces a negative result.

How the Iteration Table Is Built

Alongside the direct result, the table above shows how Newton's method converges on the same answer step by step: starting from an initial guess, each iteration refines the estimate using x_k+1 = ((n−1)·x_k + R/x_kⁿ⁻¹) / n, where R is the radicand. This is the same general approach calculators and computers have historically used internally to evaluate roots without a built-in power function, and it typically converges to full precision within just a handful of steps.

Roots, Exponents, and Related Tools

Roots and exponents are inverse operations, so if you need to go the other direction — raising a number to a power instead of extracting a root — the exponent calculator handles that. For roots and powers that aren't whole numbers, or general arithmetic on the result, the scientific calculator is also useful.