Verified calculation capability

Nth root Calculation Capability

Principal real nth root of a with positive safe-integer index n (default 2). · Math

Capability ID math.root · version 1.0.5

Open calculator REST API MCP Evidence Evidence JSON CVP Manifest Review policy

Model

nth root

nth root: ⁿ√a = a^(1/n) (a ≥ 0); ⁿ√a = −(−a)^(1/n) (a < 0, n odd); not real (a < 0, n even)
Principal value: b = ⁿ√a is the principal real value satisfying bⁿ = a (even n ⇒ b ≥ 0)
Square / cube: √a = a^(1/2) · ∛a = a^(1/3) for a ≥ 0; ∛(−a) = −∛a
Newton step: b ← [(n−1)·b + a/b^(n−1)] / n

Inputs / Outputs

Canonical symbols for this capability. Agent schemas remain the contract.

Symbol	Quantity	Unit	Role
`a`	Radicand. Share-URL alias: aInput	—	input
`aInput`	Share-URL alias for a	—	input
`n`	Root index (positive safe integer; default 2)	—	input
`nInput`	Share-URL alias for n	—	input
`index`	Alias for n	—	input
`formula`	—	—	output
`api_version`	—	—	output
`api_contract_version`	—	—	output
`spec_version`	—	—	output
`page`	Canonical calculator page URL	—	output
`links`	—	—	output

Assumptions / Validity

Declared scope for this capability. Do not use it outside this domain.

Assumptions

Real principal root only. This engine does not return complex values.
Index n is a positive safe integer (1st, square, cube, 4th, …, nth root). Generalized powers a^(1/n) with non-integer, negative, or non-safe-integer n belong on the Exponent calculator.
Even n requires a ≥ 0 and returns b ≥ 0. Odd n allows a < 0 (for example ∛(−8) = −2).
n = 2 uses Math.sqrt. n = 3 uses Math.cbrt, returning an implementation-approximated binary64 result (including negatives). Other n use a^(1/n) for a ≥ 0, or −(|a|)^(1/n) for odd n and a < 0.
The Square Root calculator is a different contract: √(x<0) returns the principal imaginary. This page rejects that case as NOT_REAL.

Example

One numerical check. The calculator page holds worked steps and FAQ.

∛27 = 3; ⁴√16 = 2; √27 ≈ 5.19615242271

Open this example in the calculator

Trust

CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance. Computational verification is defined by CVP. Supplemental domain review (internal) and named expert review are defined by the Review policy and do not by themselves determine CVP conformance.

CVP: CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Assurance: Core
Declared partition coverage: PASS · 7/7 declared partitions (sqrt-path, cbrt-path, pow-path, identity-n1, odd-negative, invalid-domain, xcal)
Calculation version: 1.0.5
Verification: 24/24 calculation tests · 10/10 legacy golden · 14/14 legacy boundary
Production surface contract: 3/3
Versions: Calculation 1.0.5 · CVP protocol 1.0.0-proposed · Evidence 2026-09-09.root-cvp
Verification revision: 2026-09-09.root-cvp · 4/4 property · digest a3f7a7bfcb95
Reference: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Numerical scope: Path-specific ULP vs O3 applies only to the 19 published tabulated vectors: identity n=1 (0 ULP); Math.sqrt including tiny/huge (≤2); Math.cbrt including odd-negative and tiny/huge (≤2); moderate Math.pow including odd-negative, ⁸√15, and max safe n (≤2); extreme Math.pow at |a|=1e±300 (≤64). It is not a whole-domain guarantee. Even roots of a<0 and illegal n are contract errors (NOT_REAL / INVALID_INDEX), not ULP claims.
Interfaces: PASS · UI (SSR) / REST / MCP / URL→result→graph
Success 1/1. Integration: URL → SSR result → Live graph current point (4/4). Hydration/slider/history are URL-canonical contracts, not a live browser session.
CVP suite: 5/5 golden · 14/14 CVP boundary · 17/17 invalid · 4/4 property · 2/2 metamorphic · 3/3 round-trip · 19/19 O3 · 1/1 cross-interface · 5/5 cross-calculator · 4/4 URL→result→graph · 1/1 CVP contract
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.0.5 (workspace artifacts agree)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

This calculator CURRENT · Public schema 1.0.5 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓

Call this capability

Human page, REST, and MCP share the same engine and calculation_version. Build verified hashes workspace artifacts only. After deploy, attest:production GETs canonical URLs twice — the public/cached view an Agent actually hits, and origin — including every /capabilities/{id} HTML page. Public Capability HTML version must match catalog before production attestation passes. Live Trust rows above fetch /.well-known/calculatorx-production-attestation.json and the output schema const. Build hashes: release integrity. That proves currently published page / engine / schema / evidence / MCP agree — not that a historical version is independently immutable. Reference Calculation Standard — Production surface contract 3/3.

Tool id: root
Engine: root
REST: POST /api/v1/calc/root
MCP: execute_calculation with capability_id: "math.root"
Agent input schema: Canonical (MCP / Agent)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.0.5)
Versioned output schema: Immutable pin (1.0.5)

References

Human-readable sources plus CVP reference evidence classes.

Sources

NIST Digital Library of Mathematical Functions, Chapter 4 — Logarithm, Exponential, Powers
NIST DLMF §4.2(iv) — Powers: principal value z^a = exp(a ln z)
CalculatorX mathematical conventions — Real nth root; even roots of negatives are NOT_REAL; n is a positive safe integer

Reference evidence classes: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior

Try with an agent

Inspect this capability via same-origin MCP, then call execute_calculation.

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: inspect_capability' \
  -d '{"jsonrpc":"2.0","id":1,"method":"tools/call","params":{"name":"inspect_capability","arguments":{"capability_id":"math.root"}}}'

Inspect the trust tier and required assurance level before execution. Pin calculation_version (1.0.5) when reproducibility matters. Full protocol notes: MCP docs.