Math calculator

Root Calculator

Q: What is an nth root?

The principal real number b such that bⁿ = a. Notation: ⁿ√a = b. For a ≥ 0 this is a^(1/n). For a < 0 and odd n it is −(−a)^(1/n), not the complex principal power. For even n, b ≥ 0, so √4 = 2 even though (−2)² = 4.

Q: What are square and cube roots?

Square root is the 2nd root (√a). Cube root is the 3rd root (∛a).

Q: Can I take an even root of a negative number?

Not in the reals — this page returns NOT_REAL for √(−4). Odd roots of negatives are real: ∛(−8) = −2. The dedicated Square Root calculator instead reports the principal imaginary (√(−9) = 3i).

Q: How is a root related to exponents?

For a ≥ 0, ⁿ√a = a^(1/n). For a < 0 and odd n, ⁿ√a = −(−a)^(1/n) so the result stays the real odd root (∛(−8) = −2), not the complex principal value of (−8)^(1/3). Raising the principal root to n recovers a: (ⁿ√a)ⁿ = a (when defined). The converse is not automatic for even n: bⁿ = a does not imply b = ⁿ√a unless b ≥ 0.

Q: How can I estimate a root by hand?

Guess b, compute c = a / b^(n−1), then replace b with [(n−1)·b + c] / n. Repeat until stable (Newton / Babylonian method).

Q: Is this the same as roots of an equation?

No. Here “root” means radical (nth root of a number). Roots of an equation are values that make it zero — use the Quadratic calculator for ax²+bx+c=0.

Q: What about n = 0, fractions, or negative n?

n must be a positive safe integer: 1 ≤ n ≤ Number.MAX_SAFE_INTEGER (2⁵³−1). n = 0 is undefined. n = 1 gives ¹√a = a. Non-integer, negative, or non-safe-integer n is not an nth-root index on this page — use the Exponent calculator for a^(1/n) in that case.

Calculate the principal real nth root ⁿ√a. Fast local calculation with shareable, machine-readable results.

Instant result

Result

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Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance

Input interpretation: Enter values to calculate.
Result: —
Assurance: Core
Declared partition coverage: PASS · 7/7 declared partitions (sqrt-path, cbrt-path, pow-path, identity-n1, odd-negative, invalid-domain, xcal) · Matrix
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.
Known limitations: Positive safe-integer n; even roots of a<0 are NOT_REAL; odd-negative uses the real branch. Declared partitions cover identity n=1, Math.sqrt, Math.cbrt, Math.pow, odd-negative, invalid-domain, and XCAL vs Square Root / Exponent — not a whole-domain claim.
Core CVP does not include live graph, viewport, or pointer interaction.
Model: The principal real nth root b = ⁿ√a. For a ≥ 0, b = a^(1/n). For a < 0 and odd n, b = −(−a)^(1/n). Default index is n = 2 when omitted.
Scope: Real principal root only. This engine does not return complex values.
Verification: Engine tested · Source checked · v1.0.5 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
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
Legacy regression: 24/24 tests · Production surface contract 3/3
Reference: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
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.
Supplemental domain review: Not performed
Named expert review: Not performed
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 · Manifest
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
Sources
Evidence: 10 legacy golden · 14 legacy boundary · legacy regression suite · 5/5 oracle-backed golden · 17/17 invalid · Artifact integrity PASS
Semantic contract: PASS

Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

nth rootⁿ√a = a^(1/n) (a ≥ 0); ⁿ√a = −(−a)^(1/n) (a < 0, n odd); not real (a < 0, n even)

Principal valueb = 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 stepb ← [(n−1)·b + a/b^(n−1)] / n

iEven n requires b ≥ 0, so √4 = 2 (not −2) even though (−2)² = 4. Even roots of negative a are not real. Odd roots of negatives use −(−a)^(1/n), not the complex principal power (e.g. ∛(−8) = −2). n must be a positive safe integer: 1 ≤ n ≤ Number.MAX_SAFE_INTEGER (2⁵³−1). ⁰√a is undefined. For equation roots (zeros of a polynomial), use the Quadratic calculator.

How to use

Choose square, cube, or nth root

For nth root, enter the index n as a positive safe integer (1 ≤ n ≤ Number.MAX_SAFE_INTEGER). Share URLs always include a and n.

Enter the radicand a

Negatives are allowed for odd integer roots only. Even roots of a < 0 are not real on this page.

Read the value and steps

See a^(1/n), a decimal approximation, and optional Newton iterations. REST and shared URLs use the same engine.

Example calculations

Common configurations with formula and result.

Square root

√16

16^(1/2)

Estimate √27

to 3 decimals

Babylonian / Newton

≈ 5.196

Cube root

∛27

27^(1/3)

Odd root of negative

∛(−8)

−2³ = −8

−2

8th root

⁸√15

15^(1/8)

≈ 1.403

Quick checks

Common values at a glance.

Expression	Result
√16	4
√27 ≈	5.196
∛27	3
∛(−8)	−2
⁴√16	2
⁸√15 ≈	1.403

i √27 ≈ 5.196152…; ⁸√15 ≈ 1.402850552…

Root calculator specification

Version 1.0.5 · Engine tested

Calculation status

Engine tested 24/24 tests · Production surface contract 3/3
Named expert review Not performed
Calculation version 1.0.5

Review policy · Evidence

Definition: The nth root of a number a is the principal real number b such that bⁿ = a, written ⁿ√a = b. The index n is a positive safe integer. For a ≥ 0, b = a^(1/n). For a < 0 and odd n, b = −(−a)^(1/n). Square root is n = 2; cube root is n = 3. For even n, b ≥ 0.
What it calculates: The principal real nth root b = ⁿ√a. For a ≥ 0, b = a^(1/n). For a < 0 and odd n, b = −(−a)^(1/n). Default index is n = 2 when omitted.
Inputs: Radicand a (finite real)
Index n (positive safe integer n ≥ 1; maximum Number.MAX_SAFE_INTEGER = 9007199254740991; default 2)
Outputs: Principal real root b
Formula: ⁿ√a = a^(1/n) (a ≥ 0); ⁿ√a = −(−a)^(1/n) (a < 0, n odd); not real (a < 0, n even)
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.
Units: This calculator operates on numeric scalar values and does not perform unit conversion. If the radicand represents a dimensioned quantity with unit U, the resulting dimension is U^(1/n).
Boundary conditions: Missing a returns MISSING_REQUIRED_INPUT. Missing n defaults to 2.
Non-numeric, object, array, boolean, NaN, or Infinity inputs return INVALID_NUMBER.
n that is not a positive safe integer (including n = 0, n < 0, non-integers such as 2.5, and n outside JavaScript's safe-integer range) returns INVALID_INDEX.
a < 0 with even n returns NOT_REAL.
a = 0 returns 0.
Defensive guards exist for unexpected non-finite numerical outcomes (RESULT_OVERFLOW / RESULT_UNDERFLOW). Overflow or underflow is not expected for valid finite nth-root inputs.
Numerical precision: Computation uses IEEE-754 binary64 (JavaScript Number): Math.sqrt for n=2, Math.cbrt for n=3, otherwise Math.pow(a, 1/n) or −Math.pow(−a, 1/n) for odd n and a < 0. Math.cbrt is implementation-approximated; ECMAScript does not require correctly rounded cbrt.
REST and SSR return that engine number. The on-page result may round for display (up to 12 significant digits; scientific notation when |b| ≥ 1e12 or 0 < |b| < 1e-6).
Values exact in binary64 (for example ⁴√16 = 2 and ∛(−8) = −2) display exactly. Irrational results such as √27 are rounded on the page; the API keeps the full binary64 value.
Interactive evaluation runs in the browser. Shared URLs and REST use the same engine server-side. Newton / Babylonian steps are UI explanations only.
The Result Card includes an engine-linked Live graph of y = x^(1/n) using the same Overview / Focus template as Log (even n: x ≥ 0; odd n: all real x). Overview always includes the current point.
Example: ∛27 = 3; ⁴√16 = 2; √27 ≈ 5.19615242271
Validation cases: 15 published on this page · 24/24 tests · Production surface contract 3/3 · View evidence
a=16, n=2 → 4
a=27, n=2 → ≈5.19615242271
a=27, n=3 → 3
a=-8, n=3 → −2
a=16, n=4 → 2
a=15, n=8 → ≈1.40285055201
a=0, n=2 → 0
a=42, n=1 → 42
a=-9, n=2 → error NOT_REAL
n=0 → error INVALID_INDEX
a=16, n=2.5 → error INVALID_INDEX
a=16, n=-2 → error INVALID_INDEX
a=16, n=9007199254740991 → ≈1
a=16, n=9007199254740992 → error INVALID_INDEX
empty a → error MISSING_REQUIRED_INPUT
Sources: NIST Digital Library of Mathematical Functions, Chapter 4 — Logarithm, Exponential, Powers
Supports: nth root as the principal real value of a^(1/n) when it exists
NIST DLMF §4.2(iv) — Powers: principal value z^a = exp(a ln z)
Supports: Principal a^(1/n) for a > 0. Negative a is restricted here to odd integer n so the result stays the real odd root, not the complex principal power.
CalculatorX mathematical conventions — Real nth root; even roots of negatives are NOT_REAL; n is a positive safe integer
Supports: Distinct from the Square Root calculator’s principal-imaginary contract for x < 0. Polynomial roots belong on Quadratic. Non-integer or non-safe-integer n is a generalized power (Exponent), not this nth-root contract. Negative odd roots use −(−a)^(1/n).
Calculation version: 1.0.5

Background

Interpretation and common distinctions.

What is a root (radical)?

The nth root of a number a is the principal real number b such that b^n = a, with

ⁿ√(a) = a^(1/n) if a ≥ 0; −(−a)^(1/n) if a < 0, n odd; not real if a < 0, n even

The index n is a positive safe integer. For even n, b ≥ 0, so √(4) = 2 even though (−2)² = 4. For a negative radicand and odd n, this engine uses −(|a|)^(1/n) so ³√(−8) = −2, not the complex principal value of (−8)^(1/3).

Name	Index n	Notation
Square root	2	√(a)
Cube root	3	³√(a)
General	n ≥ 1	ⁿ√(a)

Estimating by hand (Newton)

To approximate ⁿ√(a):

Guess b.
Compute c = a / bⁿ⁻¹.
Set b ← [(n−1) b + c] / n.
Repeat until the desired decimals stabilize.

Graph sketch

Plotting this calculator’s radicand a ≥ 0 against the principal root for a fixed index (default square root, n = 2):

b = a^(1/n) (a ≥ 0); b = −(−a)^(1/n) (a < 0, n odd); not real (a < 0, n even)

At a = 0, b = 0
For n = 2, the graph is the usual √(a) curve in the first quadrant
Negative a is omitted unless n is an odd integer (then b < 0)

For the dedicated principal square root, including √(x<0) = √(|x|) i, use the Square Root calculator.

CVP (Calculator Verification Protocol)

Math graph pilot under CVP 1.0 Proposed. Profile: core (unitless principal nth root, not an engineering model).

Evidence Manifest: /evidence/math.root/1.0.5.cvp.json
Reproduce / O3 table: /evidence/math.root/reproduce.json · /developers/cvp/reproduce/root-o3-tables.json
Independent reference: O1 (NIST DLMF model) + dedicated O3 mpmath/MPFR expected values and numerical behavior + O2 live identities
Numerical policy: IEEE-754 binary64; path-specific ULP vs tabulated O3 (declared_before_evaluation=true). Identity n=1 is 0 ULP; Math.sqrt / Math.cbrt / moderate Math.pow ≤2 ULP; extreme Math.pow at |a|=1e±300 ≤64 ULP.
Interfaces: UI (SSR), REST, and MCP must agree. A separate integration check covers URL → result → Live graph. ui-ssr is not a live browser session.
Cross-calculator: n=2 on a≥0 agrees with Square Root; even-negative is a documented contract split (NOT_REAL vs principal imaginary), not a numeric identity.
Expert review is optional and is not required for CVP Verified.
Public claim: CalculatorX CVP Verified · Core · Proposed 1.0 · Evidence available. That is CalculatorX's own CVP run plus a published independent O3 table. It is not expert review, not a third-party certification, and not a claim that a reviewer has independently reproduced the suite.

Other calculators in this family: Antilog Calculator, Exponent Calculator, Exponential Growth Calculator, Log Calculator, Natural Log (ln) Calculator, Square Root Calculator . Explore all Powers, Roots & Logarithms.

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Frequently asked questions

Key distinctions behind the calculation.

What is an nth root?