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b) Find all cube roots of 8i.

\(\begin{array}{|rcll|} \hline && \sqrt[3]{8i} \\ &=& \sqrt[3]{8}\cdot \sqrt[3]{i} \\ &=& \sqrt[3]{2^3}\cdot \sqrt[3]{i} \\ &=& 2\cdot \sqrt[3]{i} \\ \hline \end{array}\)

 

\(\begin{array}{rcll} z= a+bi &=& i \\ |z| &=& \sqrt{0^2+1^2} \\ &=& \sqrt{1} \\ &=& 1 \\ \sin(\varphi) &=& \frac{b}{|z|} \\ &=& \frac{1}{1} \\ &=& 1 \\ \varphi &=& \arcsin(1) +2k\pi \\ \varphi &=& \frac{\pi}{2} +2k\pi \\\\ z &=& |z|e^{i\varphi} \\ & = & 1\cdot e^{i\cdot ( \frac{\pi}{2}+2k\pi) } \\ \mathbf{z} & \mathbf{=} & \mathbf{ e^{i\cdot ( \frac{\pi}{2}+2k\pi) }} \\ \end{array}\)

 

\(\begin{array}{|rcll|} \hline \sqrt[3]{i} \\ &=& (i)^{\frac13} \\ &=& \left( e^{i\cdot ( \frac{\pi}{2}+2k\pi)}\right)^{\frac13} \\ &=& e^{i\cdot \left( \frac{\frac{\pi}{2}+2k\pi}{3} \right)} \\ &=& e^{i\cdot \left( \frac{\pi}{6}+\frac{2}{3}\cdot k\pi \right) } \quad & | \quad k =(0,1,2) \\\\ \sqrt[3]{i} &=& e^{i\cdot \left( \frac{\pi}{6} \right)} \quad & | \quad k = 0 \\ &=& \cos( \frac{\pi}{6} ) + i \cdot \sin(\frac{\pi}{6} ) \quad & | \quad \frac{\pi}{6} = 30^{\circ}\\ &=& \cos( 30^{\circ} ) + i \cdot \sin(30^{\circ}) \\ &=& \frac{\sqrt{3}}{2} + i\cdot \frac12 \\ 2\cdot \sqrt[3]{i} &=& 2\cdot \left( \frac{\sqrt{3}}{2} + i\cdot \frac12 \right) \\ \mathbf{2\cdot \sqrt[3]{i}} &\mathbf{=}& \mathbf{\sqrt{3} + i} \\\\ \sqrt[3]{i} &=& e^{i\cdot \left( \frac{\pi}{6}+\frac23 \pi \right)} \quad & | \quad k = 1 \\ &=& e^{i\cdot \left( \frac{5\pi}{6} \right)} \\ &=& \cos(\frac{5\pi}{6}) + i \cdot \sin(\frac{5\pi}{6}) \quad & | \quad \frac{5\pi}{6} = 150^{\circ}\\ &=& \cos( 150^{\circ} ) + i \cdot \sin(150^{\circ}) \\ &=& -\cos( 30^{\circ} ) + i \cdot \sin(30^{\circ}) \\ &=& -\frac{\sqrt{3}}{2} + i\cdot \frac12 \\ 2\cdot \sqrt[3]{i} &=& 2\cdot \left( -\frac{\sqrt{3}}{2} + i\cdot \frac12 \right) \\ \mathbf{2\cdot \sqrt[3]{i}} &\mathbf{=}& \mathbf{-\sqrt{3} + i} \\\\ \sqrt[3]{i} &=& e^{i\cdot \left( \frac{\pi}{6}+2\cdot \frac23 \pi \right)} \quad & | \quad k = 2 \\ &=& e^{i\cdot \left( \frac{3\pi}{2} \right)} \\ &=& \cos(\frac{3\pi}{2}) + i \cdot \sin(\frac{3\pi}{2}) \quad & | \quad \frac{3\pi}{2} = 270^{\circ}\\ &=& \cos( 270^{\circ} ) + i \cdot \sin(270^{\circ}) \\ &=& 0 + i \cdot \sin(90^{\circ}) \\ &=& -i \\ 2\cdot \sqrt[3]{i} &=& 2\cdot( -i ) \\ \mathbf{2\cdot \sqrt[3]{i}} &\mathbf{=}& \mathbf{-2i} \\\\ \hline \end{array}\)


All 3rd roots of \(8 i\) :
\(\sqrt{3} + i \\ -\sqrt{3} + i \\ -2i\)

 

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heureka 12.12.2017