Worked example
Square root
- Input
- n = 144; k = 2
- Output
- r = 12
Twelve squared returns the original input n.
Radical Calculator
Simplify an integer square, cube, or higher root by extracting complete powers, and also report its decimal approximation. Negative radicands are accepted only with odd indexes. This page covers “under root 20”.
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How to use this calculator
Enter an integer radicand and a whole-number root index from 2 through 20.
Review real versus complex domain, radicand sign, whole-number index range, and displayed decimal rounding.
Read exact simplified form, decimal approximation, outside factor, and remaining radicand.
Formula
Prime exponents are divided by the root index: each quotient moves outside the radical and each remainder stays inside. The approximation is x^(1/n) in the real domain.
Worked example
The square root of 72 simplifies exactly to 6√2 and is approximately 8.4852813742.
Reference guide
Formula details and result guidance
Follow the variables, derivation, precision rule, and worked cases below to reproduce the calculation independently.
Variables
| Symbol | Meaning | Unit | Role |
|---|---|---|---|
| n | Number under the root sign | none | Input |
| k | Root index | none | Input |
| r | Principal real root | none | Output |
Derivation
- The principal root with index k, r, is defined by rᵏ = n.
- For positive n, r = exp(ln(n) ÷ k); when k is odd, negative n is also allowed and returns a negative real result.
- When k is even and n is negative, there is no real-valued result.
Rounding and precision
Results that are exact integer powers are normalized when identifiable; otherwise the displayed decimal uses up to 10 places. The page reports a real principal root, not the full set of complex roots.
Worked and boundary examples
Worked example
Odd root of a negative
- Input
- n = −27; k = 3
- Output
- r = −3
An odd power preserves the negative sign.
Boundary case
Zero input
- Input
- n = 0; k = 7
- Output
- r = 0
Zero has a real root of zero for every positive integer index k.
Common mistakes
- Expecting a real result for an even root of a negative number.
- Confusing the principal square root √n with both solutions to the equation x² = n.
- Rounding an irrational root before using it in a later calculation.
How to interpret the result
The output is the principal real number whose power with index k equals the input n. A displayed decimal may be an approximation even when the underlying relationship is exact.
Scope
Appropriate for
- Evaluating square roots and higher real roots.
- Checking a candidate root by raising it back to the selected index.
Do not use for
- Enumerating all complex roots of a number.
- Treating a rounded decimal as an exact symbolic simplification.
Tools for the same task
Next tools in the workflow
Evidence and sources
- A principal root with index k reverses the corresponding power with index k on its real domain.OpenStax: College Algebra 2e — Preface and ScopeSupports: Slope, equations, radicals, proportions, and inverse algebraic operations. Reviewed 2026-08-22.
- Negative inputs have real odd-index roots but no real even-index roots.OpenStax: Prealgebra 2e — Preface and ScopeSupports: Percent, proportion, radical, ratio, and elementary geometry procedures. Reviewed 2026-08-22.
Assumptions and limits
- Exact factor extraction is limited to safe integer radicands; complex roots and symbolic expressions are not supported.
- Radical Calculator treats the entered values as estimates and does not infer missing project, account, or personal details.
- Calculations retain working precision except where the cited method requires an intermediate rounding step; displayed values are then formatted separately. Source rules and dated rates must be rechecked when they change.
Sources
- OpenStax: College Algebra 2eAlgebraic, ratio, radical, equation, and function formulas. Reviewed 2026-08-14.
- OpenStax: Elementary Algebra 2eOfficial formula or method source used by this calculator. Reviewed 2026-08-14.
Related calculators
Verification
How this calculator is checked
Trust comes from reproducible evidence, not model confidence. You can inspect the formula, assumptions, worked example, and cited sources on this page.
- Versioned calculationFormula logic is kept separate from the interface and covered by automated registry and behavior checks.
- Cited evidenceSources linked for independent checking: 2.
- Transparent AI useAI may assist drafting or adversarial review. Agreement between models is not proof, and no human expert review is claimed unless a named reviewer is shown.
Frequently asked questions
How does the Radical Calculator work?
Prime exponents are divided by the root index: each quotient moves outside the radical and each remainder stays inside. The approximation is x^(1/n) in the real domain.
What information should I enter in the Radical Calculator?
Enter an integer radicand and a whole-number root index from 2 through 20, then verify real versus complex domain, radicand sign, whole-number index range, and displayed decimal rounding.
What should I verify before using this Radical Calculator result?
Exact factor extraction is limited to safe integer radicands; complex roots and symbolic expressions are not supported.
Last updated 2026-08-25 · integer-factorization-v2 · content-2026-08-14 · Published by YunFanLabs