Find all cube roots of -2+2i.

Question

Find all cube roots of -2+2i.

Answer 1

Find all cube roots of -2+2i.

re^iθ = -2+2i = 2 (-1 + i)  = 2√2 ( cos3π/4 + i sin3π/4) = 2√2e^i3π/4

Find cube roots.

(re^iθ)^1/3 = (2√2e^i3π/4)1/3 = (2√2)^1/3 e^iπ/4= (2√2)^1/3 ( cos(3π/4 + 2kπ)/3+ i sin(3π/4 + 2kπ)/3)

The three cube roots are obtained by setting k = 0,1,2

 

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