Apply De Moivre's Theorem to raise complex numbers to any power and find all nth roots.
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Examples
Examples
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All n roots:
De Moivre's Theorem
(r·cis θ)ⁿ = rⁿ·cis(nθ)
To raise a complex number to the nth power:
Step 1: Convert z = a + bi to polar form: r = √(a² + b²), θ = atan2(b, a).
Step 2: Raise the modulus to the nth power: rⁿ.
Step 3: Multiply the argument by n: nθ.
Step 4: Convert back to rectangular: rⁿ·cos(nθ) + i·rⁿ·sin(nθ).
Nth Roots of a Complex Number
wₖ = r^(1/n)·cis((θ + 2πk)/n), k = 0,1,…,n−1
Every nonzero complex number has exactly n distinct nth roots, equally spaced around a circle of radius r^(1/n).
Step 1: Convert z to polar: r, θ.
Step 2: Compute the root modulus: r^(1/n).
Step 3: For each k = 0, 1, …, n−1 compute angle (θ + 2πk)/n.
The roots are separated by 2π/n radians (360°/n) on the Argand diagram.
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