Root Calculator

How to use?

Pick the root type, type in the radicand and the result shows up. Keep in mind that the n-th root of x is the same as x^(1/n). For even-index roots, the radicand has to be positive.

Roots: definition and Newton-Raphson

The square root is the inverse of squaring: √x = x^(1/2). More generally, the n-th root is ⁿ√x = x^(1/n). Modern calculators compute square roots with the Newton-Raphson method: starting from a guess x₀, iterate x_{n+1} = (x_n + a/x_n)/2 — it converges quadratically (number of correct digits roughly doubles each step). For even n, the radicand must be x ≥ 0 in ℝ; in ℂ, negative radicands have complex roots. Note the difference between the principal root and all roots: the equation x² = 4 has two solutions (±2), but √4 denotes only the principal (positive) root, 2. The Babylonians computed √2 ≈ 1.41421 on clay tablets around 1800 BCE.

Applications

Square roots appear in Euclidean distance (d = √((x₂-x₁)² + (y₂-y₁)²)), the Pythagorean theorem for hypotenuse, RMS (root mean square — used in AC voltage and audio), standard deviation in statistics (square root of variance), and many geometry formulas (diagonal of a square, radius from area). Cube roots arise in volume-to-edge conversions and in solving cubic equations (Cardano).

FAQ

Why can't I take √(-1) on a calculator? Real-number calculators reject negative radicands for even indices. The result is the imaginary unit i, defined in complex numbers.

Is √4 equal to ±2? No. By convention, the radical symbol denotes the principal (non-negative) root: √4 = 2. The equation x² = 4 has two solutions, but √4 alone is just 2.

Can odd roots accept negative numbers? Yes. ³√(-8) = -2 because (-2)³ = -8. Only even-index roots restrict the domain.

What is ⁿ√x when n is fractional? Roots and powers are unified: x^(1/n) generalizes to x^r for any real r via the exponential function.