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Math calculator

Complex Power Calculator

Integer power of a complex number via De Moivre. Same engine as Complex Number. Not a CAS. Not i^i. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 18/18 tests · Production surface contract 3/3 · v1.1.0
Input interpretation
Enter values to calculate.
Result
—
Model
Integer power z^n of one complex number.
Scope
Integer n
Verification
Engine tested · 18/18 tests · Production surface contract 3/3 · Specification checked · v1.1.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
6 golden · 4 boundary · 8 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: STALE · Last attested schema matched 1.1.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

De Moivrez^n = r^n (cos nθ + i sin nθ)
Negative nz^{−n} = 1 / z^n
iInteger n, |n| ≤ 1000. Float64. 0^0 is undefined. Not i^i (non-integer / complex exponent). Not a second complex engine.

How to use

1

Enter re, im, and n

n is an integer. Negative n inverts. Larger composition lives at /calc/math/complex-number.

2

Read zⁿ

Every result includes rectangular, polar, and Euler form.

Example calculations

Common configurations with formula and result.

ϟ

i²

0+1i to the 2

i² = −1
−1
ϟ

(3+4i)²

De Moivre

−7+24i
−7 + 24i

Complex Power calculator specification

Version 1.1.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
z^n = r^n (cos nθ + i sin nθ) for integer n. This page is mode=power on math.complex. Nth roots and Euler form share the same engine. Not the real exponent page.
What it calculates
Integer power z^n of one complex number.
Inputs
  • re
  • im
  • n
Outputs
  • re
  • im
  • r
  • arg_deg
  • euler
Formula
z^n = r^n cis(nθ)
Assumptions
  • Integer n
  • Float64
  • Not i^i
  • Not a CAS
Units
  • dimensionless
  • degrees
Boundary conditions
  • 0^0 → INVALID_INPUT
  • 0^{n<0} → DIVISION_BY_ZERO
  • |n|>1000 → VALUE_ABOVE_MAX
Example
re=0 im=1 n=2 → −1
Validation cases

2 published on this page · 18/18 tests · Production surface contract 3/3 · View evidence

  • re=0 im=1 n=2 → re=-1 im=0
  • re=3 im=4 n=2 → re=-7 im=24
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.1.0

Other calculators in this family: Complex Number Calculator, Complex Polar Calculator, Complex Roots Calculator, Euler Form Calculator . Explore all Complex Numbers.

→ Complex Number Calculator Add, subtract, multiply, divide complex numbers; convert rectangular ↔ polar; integer powers, nth roots, and Euler form. Not a CAS. Runs locally. › → Complex Polar Calculator Convert a complex number between rectangular (re, im) and polar (r, θ°) form. Same engine as Complex Number. Not a CAS. Runs locally. › → Complex Roots Calculator All n nth-roots of a complex number via De Moivre. Same engine as Complex Number. Not a CAS. Not the real root page. Runs locally. › → Euler Form Calculator Euler form of a complex number: z = r e^{iθ} = r (cos θ + i sin θ). Same engine as Complex Number. Not a CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second complex engine?

No. It is mode=power on math.complex, the same engine as /calc/math/complex-number.

Is i^i supported?

No. This seed is integer n only. Complex exponents stay out of scope.