find all the roots to the equation: z^3 = 1 + i in polar form
1st write 1+ i in polar form, sketch a diagram to find the angle ( =pi/4) and find the modulus ( sqrt(2))z^3 = Sqrt(2) E^ipi/4This is true for all equivalent solutions (add 2kpi) z^3 = sqrt(2)e^(pi/4 +2kpi)iUse De moivres theorem: z = 2^(1/6) e^(pi/12 +2kpi/3)iThis is an algebraic equation, so has 3 solutions (since z^3 is the highest power) answers are usually given with angles in range -pi < x < pi. So our solutions correspond to k = 0, 1, -1z = 2^1/6 e^ipi/12 , 2^1/6 e^i3pi/4 , 2^1/6 e^-i7pi/12
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Answered by Rajan M. • Further Mathematics tutor
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