Radical (Root) Calculator

A professional-grade root calculator to compute n-th roots accurately, simplify radicals, detect perfect powers, and present real or complex results with clear steps. Ideal for students, engineers, and educators who need trustworthy, accessible, and fast results.

Root Calculator Inputs

Choose number domain

Results

Simplified Radical (exact) —
Decimal Approximation —
Perfect n-th Power? —
Principal Complex Root —

Steps

    Authoritative Content

    Data Source and Methodology

    Primary reference: NIST Digital Library of Mathematical Functions (DLMF), Chapter 1: Elementary Transcendental Functions (Release 1.1.9 of 2023-06-15). Direct link: https://dlmf.nist.gov/1. All derivations and properties (principal roots, polar form, and extraction of perfect powers) are consistent with standard mathematical conventions.

    Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.

    The Formula Explained

    For x ≥ 0 and integer n ≥ 2, the principal n-th root is defined by:

    $$ y = \sqrt[n]{x} \iff y^n = x,\quad y \ge 0. $$

    Radical simplification (for x ∈ ℕ) uses prime factorization. If $$ x = \prod_{i} p_i^{\,\alpha_i}, $$ then $$ \sqrt[n]{x} = \left( \prod_{i} p_i^{\,\left\lfloor \alpha_i/n \right\rfloor} \right)\cdot \sqrt[n]{\prod_{i} p_i^{\,\alpha_i \bmod n}}. $$

    For complex results (principal root). If $$ z = r\,e^{i\theta},\quad r \ge 0,\ \theta\in(-\pi,\pi], $$ then the principal n-th root is $$ z^{1/n} = r^{1/n} e^{i\theta/n} = r^{1/n}\big(\cos(\tfrac{\theta}{n}) + i\sin(\tfrac{\theta}{n})\big). $$

    All n complex roots are given by: $$ z_k = r^{1/n} e^{i(\theta + 2\pi k)/n},\quad k=0,1,\dots,n-1. $$

    Glossary of Variables

    Radicand (x): The number under the radical sign.
    Index (n): The root degree (2 = square, 3 = cube, etc.).
    Precision: Number of decimals for numeric outputs.
    Simplified Radical: Exact form, e.g., a·√[n]{b} with b as small as possible.
    Decimal Approximation: Rounded numeric value to your precision.
    Principal Complex Root: The k=0 root in polar/a+bi form when complex output is needed.

    How It Works: A Step-by-Step Example

    Goal: Simplify and evaluate √500 with 6 decimals.

    1. Factorize 500 = 2² × 5³.
    2. Extract perfect squares (groups of 2): 2 and 5² leave 10 outside the radical; one 5 remains inside.
    3. Result: √500 = 10√5.
    4. Decimal: √500 ≈ 22.360680.

    If you switch the domain to Complex and use an even index on a negative radicand, the tool computes the principal root using polar form and can list all n equally spaced complex roots.

    Frequently Asked Questions (FAQ)

    Is there any difference between √x and x^(1/2)?

    For real x ≥ 0, they coincide and refer to the principal square root. For negative or complex values, branch choices matter; this tool uses the principal branch.

    When can a radical be simplified?

    When the radicand is a non-negative integer, factorization can extract perfect n-th powers. For decimals or rationals, simplification may not apply in integer-factorization terms.

    What if the input is a perfect n-th power?

    The tool reports an exact integer result and indicates that the radicand is a perfect n-th power.

    How accurate are the decimals?

    Calculations use double-precision floating point. The displayed value is rounded to your selected precision (0–12 decimals).

    Do you support all complex n-th roots?

    Yes. In Complex domain, open the “All n complex roots” panel to see all roots z_k = r^{1/n} e^{i(θ+2πk)/n} for k = 0,…,n−1.

    Why do I get an error for negative radicands with even indices in Real mode?

    Because no real number raised to an even power yields a negative result. Switch to Complex mode for a valid principal complex root.

    Tool developed by Ugo Candido.
    Content reviewed by the CalcDomain Editorial Team.
    Last reviewed for accuracy on: .