3-a-Find-the-complex-number-z-which-satisfy-the-equation-z-3-8i-giving-your-answer-in-the-form-a-bi-where-a-and-b-are-real-

Question Number 94080 by Rio Michael last updated on 16/May/20

3 . (a) Find the complex number z which satisfy

the equation z^{3} = 8 i, giving your answer in the

form a + b i where a and b are real .

Answered by mr W last updated on 17/May/20

s a y z = {r e}^{i θ} = r (\cos θ + i \sin θ)

r^{3} e^{3 θ i} = 8 i = 2^{3} e^{\frac{π}{2} i}

\Rightarrow r = 2

\Rightarrow 3 θ = \frac{π}{2} + 2 k π \Rightarrow θ = \frac{π}{6} + \frac{2 k π}{3} w i t h k = 0, 1, 2

\Rightarrow z = 2 e^{(\frac{π}{6} + \frac{2 k π}{3}) i} = 2 [\cos (\frac{π}{6} + \frac{2 k π}{3}) + i \sin (\frac{π}{6} + \frac{2 k π}{3})]

= \sqrt{3} + i, - \sqrt{3} + i, - 2 i

Commented by Rio Michael last updated on 17/May/20

thank you sir .