Complex Powers (De Moivre)
Powers and roots of complex numbers
About this calculator
The Complex Powers Calculator raises a complex number z = a + bi to any integer power up to ±100, or finds all n distinct nth roots (up to 24), using De Moivre's theorem. Integer powers use exact binary exponentiation so Gaussian-integer inputs stay exact; roots come from the polar form ρ·e^{i(θ+2πk)/n} at 30-digit precision. Every result is shown in both rectangular and polar form with modulus and argument, and 0 raised to a non-positive power is reported as undefined.
How to use the Complex Powers (De Moivre) calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
Common examples
- (1 + i)⁴ → −4 exactly (modulus 4, argument 180°)
- i² → −1, the defining identity of the imaginary unit
- Cube roots of 8 → 2, −1 + 1.732051i, −1 − 1.732051i, spaced 120° apart
- Fourth roots of 1 → 1, i, −1, −i on the unit circle
- 2⁻² → 0.25 via the reciprocal of the positive power
Frequently asked questions
How does the Complex Powers (De Moivre) calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Complex Powers (De Moivre) calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.