find-the-zero-of-z-3-729-0-z-C-

Question Number 142623 by Engr_Jidda last updated on 03/Jun/21

f i n d t h e z e r o o f z^{3} + 729 = 0

z \in C

Answered by Olaf_Thorendsen last updated on 03/Jun/21

z^{3} = - 729 = 9^{3} e^{i π}

z = 9 e^{i (\frac{π}{3} + \frac{2 k π}{3})}, k = 0, 1, 2

z \in {\frac{9}{2} (1 \pm i \sqrt{3}), - 9}

Answered by Rasheed.Sindhi last updated on 03/Jun/21

z^{3} + 9^{3} = 0

(z + 9) (z^{2} - 9 z + 81) = 0

z + 9 = 0 ∣ z^{2} - 9 z + 81 = 0

z = - 9 ∣ z = \frac{9 \pm \sqrt{81 - 4 (81)}}{2}

∣ z = \frac{9 \pm \sqrt{- 3 (81)}}{2}

∣ z = \frac{9 \pm 9 i \sqrt{3}}{2} = - 9 (\frac{- 1 \mp i \sqrt{3}}{2})

∣ z = - 9 ω, - 9 ω^{2}

z = - 9, - 9 ω, - 9 ω^{2}