# Nth Root Calculator

> Computes the principal n-th root of a number (value^(1/n)), returning the real negative root for odd n and the second, negative root for even n.

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

## Purpose

Computes the principal n-th root of a number (value^(1/n)), returning the real negative root for odd n and the second, negative root for even n.

**Use when:** You need √x, ∛x or any integer-degree root, including odd roots of negative numbers.

**Do not use when:** You need a fractional power such as x^(2/3) (use exponent) or the solutions of a quadratic (use quadratic-equation).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `value` | number |  | required | The number under the root sign. Negative values are allowed only for odd n. (min -1e+300, max 1e+300) |
| `n` | integer |  | required | 2 for square root, 3 for cube root, etc. (min 1, max 1000) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `root` | number |  | The real n-th root with the same sign as value. |
| `negative_root` | number |  | −root: the second real root, present only for even n and positive value. |
| `verification` | number |  | The root raised back to the n-th power (should equal value). |
| `expression` | string |  | The root as text, e.g. '16^(1/4) = 2'. |

## Formula

`root = |value|^(1/n) × sign(value); for even n and value > 0 the roots are ±root`

Roots that are exact integers are snapped to the integer when |value|^(1/n) is within 10^-9 of it and the integer power reproduces value exactly.

## Data Sources

- Wikipedia – Nth root — https://en.wikipedia.org/wiki/Nth_root (reference, retrieved 2026-09-23)
- Wolfram MathWorld – nth Root — https://mathworld.wolfram.com/nthRoot.html (reference, retrieved 2026-09-23)

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/nth-root?value=…&n=…`
- `POST https://tttkmbb.com/api/v1/calculate/nth-root` 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/nth-root · OpenAPI operationId `calculate_nth_root` 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": "nth-root", "inputs": {…}}`

## Example

- Fourth root of 16: inputs `{"value":16,"n":4}` → `{"root":2,"negative_root":-2,"verification":16,"expression":"16^(1/4) = 2"}`
- Fifth root of −32: inputs `{"value":-32,"n":5}` → `{"root":-2,"verification":-32}`

```
GET https://tttkmbb.com/api/v1/calculate/nth-root?value=16&n=4
```

## Limitations

You need a fractional power such as x^(2/3) (use exponent) or the solutions of a quadratic (use quadratic-equation). Roots that are exact integers are snapped to the integer when |value|^(1/n) is within 10^-9 of it and the integer power reproduces value exactly. All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Why does the square root of a negative number fail?**

Even roots of negative numbers are imaginary (√−4 = 2i); only odd roots of negatives are real, e.g. ∛−27 = −3.

**Why are there two answers for an even root?**

Both 2 and −2 satisfy x⁴ = 16. The principal root is the positive one; negative_root gives the other.

## Related

- [Exponent Calculator](https://tttkmbb.com/math/exponent.md) — Inverse operation and fractional exponents.
- [Logarithm Calculator](https://tttkmbb.com/math/logarithm.md) — Roots via logarithms: log(root) = log(value) / n.
- [Quadratic Equation Solver](https://tttkmbb.com/math/quadratic-equation.md) — Roots of second-degree equations.