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Question Number 77132 by peter frank last updated on 03/Jan/20

Commented by kaivan.ahmadi last updated on 03/Jan/20

$$z=4\sqrt{3}(\frac{1}{2}+i\frac{\sqrt{3}}{2})-4(-\frac{\sqrt{3}}{2}+i\frac{1}{2})=$$$$2\sqrt{3}+6i+2\sqrt{3}-2i=4\sqrt{3}+4i=4(\sqrt{3}+i)=8(\frac{\sqrt{3}}{2}+i\frac{1}{2})=$$$$8e^{i\frac{\pi }{6}}$$$$\Rightarrow \frac{z}{8}+i{\left(\frac{z}{8}\right)}^2+{\left(\frac{z}{8}\right)}^3=e^{i\frac{\pi }{6}}+{ie}^{i\frac{\pi }{3}}+e^{i\frac{\pi }{2}}=2e^{i\frac{\pi }{2}}$$$$since$$$$e^{i\frac{\pi }{6}}+{ie}^{i\frac{\pi }{3}}=\frac{\sqrt{3}}{2}+i\frac{1}{2}+\frac{1}{2}i-\frac{\sqrt{3}}{2}=i=e^{i\frac{\pi }{2}}$$

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