Willkommen hier:
Aufgabe 1:
\( z^{3}=8 \)
allgemein gilt:
\( z_{k}=|a|^{\frac{1}{n}} \cdot e^{i \frac{\ φ+2 k \pi}{n}} \quad(k=0,1,2) \)
\( |a|=\sqrt{(\text { Realteil })^{2}+(\text {Imaginärteil })^{2}}=\sqrt{8^{2}+0^{2}}=8 \)
\( \tanφ=\frac{\text { Imaginärteil }}{\operatorname{Realteil }}=\frac{0}{8}=0 \)
\( φ=0^{\circ} \)
\( n=3 \)
\( z_{0}=8^{\frac{1}{3}} e^{i \frac{0^{°}+2 0 \pi}{3}}=2 e^{0}=2 \)
\( z_{1}=2 e^{i \frac{0+2 \cdot 1 \pi}{3}}=2 e^{i \frac{2 \pi}{3}} \)
\( z_{1}=2\left(\cos \left(\frac{2 \pi}{3}\right)+i \sin \left(\frac{2 \pi}{3}\right)\right) \)
\( z_{1}=2\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right) \)
z1 = -1 +i \( \sqrt{3} \)
\( z_{2}=2 \cdot e^{i \frac{0 +2*2\pi}{3}}=2 e^{i \frac{4 \pi}{3}} \)
\( z_{2}=2\left(\cos \left(\frac{4 \pi}{3}\right)+i \sin \left(\frac{4 \pi}{3}\right)\right) \)
\( z_{2}=2\left(\cos \left(240^{\circ}\right)-i \sin \left(240^{\circ}\right)\right) \)
\( z_{2}=2\left(-\frac{1}{2}-i\frac{\sqrt{3}}{2}\right)=-1-i \sqrt{3} \)
