# Radical Simplifier

> Writes √n or ∛n (optionally with a coefficient) in simplest radical form a√b by extracting perfect squares or cubes from the prime factorization of the radicand, and gives the decimal value.

- Calculator id: `simplify-radical` · Category: Math (`math`) · Tool name: `calculate_simplified_radical`
- Canonical page: https://tttkmbb.com/math/simplify-radical · This document: https://tttkmbb.com/math/simplify-radical.md · JSON definition: https://tttkmbb.com/math/simplify-radical.json

## Purpose

Writes √n or ∛n (optionally with a coefficient) in simplest radical form a√b by extracting perfect squares or cubes from the prime factorization of the radicand, and gives the decimal value.

**Use when:** You need √50 as 5√2, ∛54 as 3∛2, or want to know whether an integer is a perfect square or cube.

**Do not use when:** You only need the numeric value of a root (use nth-root), a root of a non-integer, or roots of degree 4 and higher.

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `radicand` | integer |  | required | Integer under the root sign (negative only with index 3). (min -1000000000000, max 1000000000000) |
| `index` | integer |  | optional, default 2 | 2 for a square root, 3 for a cube root. (min 2, max 3) |
| `coefficient` | integer |  | optional, default 1 | Optional multiplier in front of the radical, e.g. 3 for 3√50. (min -1000000, max 1000000) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `simplified` | string |  | Simplest form such as '5√2', '3∛2', '-2∛3' or a plain integer when the radicand is a perfect power. |
| `outside_coefficient` | integer |  | The integer in front of the simplified radical (including the input coefficient and sign). |
| `remaining_radicand` | integer |  | The integer left under the root sign (1 when the root is exact). |
| `decimal` | number |  | coefficient × n^(1/index). |
| `is_perfect_power` | boolean |  | true when the root is an integer. |
| `factorization` | string |  | Prime-power form of \|n\|, e.g. '50 = 2 × 5^2'. |
| `expression` | string |  | e.g. '√50 = √(25 × 2) = 5√2'. |

## Formula

`n = Π pᵢ^eᵢ; ⁿ√n = (Π pᵢ^⌊eᵢ/k⌋) · ᵏ√(Π pᵢ^(eᵢ mod k)) for index k; for a negative radicand and k = 3 the sign moves outside`

## Data Sources

- Wikipedia – Nth root – simplified form of a radical expression — https://en.wikipedia.org/wiki/Nth_root#Simplified_form_of_a_radical_expression (reference, retrieved 2026-09-24)
- Wikipedia – Square root — https://en.wikipedia.org/wiki/Square_root (reference, retrieved 2026-09-24)
- Wolfram MathWorld – Radical — https://mathworld.wolfram.com/Radical.html (reference, retrieved 2026-09-24)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/simplify-radical?radicand=…`
- `POST https://tttkmbb.com/api/v1/calculate/simplify-radical` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/simplify-radical · OpenAPI operationId `calculate_simplified_radical` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "simplify-radical", "inputs": {…}}`

## Example

- √50: inputs `{"radicand":50,"index":2,"coefficient":1}` → `{"simplified":"5√2","outside_coefficient":5,"remaining_radicand":2,"decimal":7.07106781,"is_perfect_power":false,"factorization":"50 = 2 × 5^2","expression":"√50 = √(25 × 2) = 5√2"}`
- ∛54: inputs `{"radicand":54,"index":3}` → `{"simplified":"3∛2","outside_coefficient":3,"remaining_radicand":2,"decimal":3.77976315,"is_perfect_power":false,"factorization":"54 = 2 × 3^3","expression":"∛54 = ∛(27 × 2) = 3∛2"}`

```
GET https://tttkmbb.com/api/v1/calculate/simplify-radical?radicand=50&index=2&coefficient=1
```

## Limitations

You only need the numeric value of a root (use nth-root), a root of a non-integer, or roots of degree 4 and higher. All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**What does 'simplest radical form' mean?**

The radicand contains no factor that is a perfect square (for √) or perfect cube (for ∛): √50 = √(25 × 2) = 5√2, because 25 is the largest square factor of 50.

**Why is the square root of a negative number rejected?**

√−n is imaginary (i√n), not a real radical; only odd roots of negative numbers are real, e.g. ∛−54 = −3∛2.

## Related

- [Nth Root Calculator](https://tttkmbb.com/math/nth-root.md) — Numeric value of any root, including higher indices.
- [Prime Factorization Calculator](https://tttkmbb.com/math/prime-factorization.md) — The factorization used to extract perfect powers.
- [Quadratic Equation Solver](https://tttkmbb.com/math/quadratic-equation.md) — Roots that often need to be simplified.