Root Calculator
Calculate the square root, cube root or nth root of any number. Result with 10 decimal places of precision.
Verification
Exponential notation
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.
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Calculate roots of any index
Everyone can pull off a square root, more or less. But the cube root of a number? Or a root of any index at all? That's usually where the standard calculator button gives up. This one keeps going and extracts the square, cube or nth root of any value, accurate to ten decimal places.
Using it is simple. You enter the number and the index of the root, and the result follows right away. It helps with maths exercises, engineering calculations, statistics, and any situation where the root isn't the usual square one, which happens to be the case an ordinary calculator can't always handle.
The whole calculation happens in the browser, with the precision that serious work tends to demand in the decimal places. It's a direct tool for when the math goes past the trivial, with nothing to install.