How to calculate the nth root of a complex number?

To calculate the nth root of a complex number, use De Moivre's Theorem and polar form.

De Moivre's Theorem states that for any complex number z = r(cosθ + i sinθ), the nth root of z is given by:

z^(1/n) = r^(1/n) [cos(θ/n + 2πk/n) + i sin(θ/n + 2πk/n)]

where k = 0, 1, 2, ..., n-1.

To find the nth root of a complex number in rectangular form, first convert it to polar form using:

r = |z| = √(x^2 + y^2)

θ = arg(z) = tan^(-1)(y/x)

where x and y are the real and imaginary parts of z, respectively.

Then apply De Moivre's Theorem to find the nth root.

Example: Find the cube roots of z = 8 + 8i.

First, convert z to polar form:

r = |z| = √(8^2 + 8^2) = 8√2

θ = arg(z) = tan^(-1)(8/8) = π/4

So z = 8√2(cos(π/4) + i sin(π/4)).

Using De Moivre's Theorem, the cube roots of z are:

z^(1/3) = (8√2)^(1/3) [cos(π/12), + i sin(π/12)]

z^(1/3) = (8√2)^(1/3) [cos(5π/12), + i sin(5π/12)]

z^(1/3) = (8√2)^(1/3) [cos(9π/12), + i sin(9π/12)]

Simplifying, we get:

z^(1/3) = 2(cos(π/12) + i sin(π/12))

z^(1/3) = 2(cos(5π/12) + i sin(5π/12))

z^(1/3) = 2(cos(3π/4) + i sin(3π/4))