Question

Finding the nth Roots of a Complex Number: Find the three cube roots of z=-2+2i

Answer 1

To find the three cube roots of the complex number z = -2 + 2i, use De Moivre's Theorem to convert the number to polar form and then calculate the cube roots using the formula.

Step-by-step explanation:

To find the three cube roots of the complex number z = -2 + 2i, we can use De Moivre's Theorem. First, we can write the complex number in polar form as r(cosθ + isinθ), where r is the modulus (distance from the origin) and θ is the argument (angle from the positive real axis).

For z = -2 + 2i, r = √((-2)^2 + 2^2) = 2√2, and θ = arctan(2/(-2)) = -pi/4.

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

• ∛z₁ = ∛(2√2)(cos(-pi/12) + isin(-pi/12))
• ∛z₂ = ∛(2√2)(cos(-5pi/12) + isin(-5pi/12))
• ∛z₃ = ∛(2√2)(cos(-9pi/12) + isin(-9pi/12))