Powers and roots of complex numbers

2000002109

Level: 
A
Given \( z= \sqrt[3]{3} \left(\cos{\frac{3\pi}{4}}+i\sin{\frac{3\pi}{4}}\right) \), determine which of the numbers below does not represent \( z^6\).
\( 9 \)
\( 9i \)
\( 9\left(\cos{\frac{9\pi}{2}}+i\sin{\frac{9\pi}{2}}\right) \)
\( 9\left(\cos{\frac{\pi}{2}}+i\sin{\frac{\pi}{2}}\right) \)

2000002108

Level: 
A
Find the value of the principal argument \( \varphi\) of \( \left(3\left(\cos{\frac{3\pi}{2} }+ i\sin{\frac{3\pi}{2} }\right)\right)^{13} \). The principal value is the corresponding angle \(\varphi \in (-\pi; \pi] \).
\( -\frac{\pi}{2} \)
\( \frac{\pi}{2} \)
\( 0 \)
\( \frac{3}{26}\pi \)

2000002102

Level: 
A
Consider \( z= \cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}} \) and find \(z^9\).
\( \cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}} \)
\( 9 \left(\cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}}\right) \)
\( \cos{\frac{9\pi}{4}} - i\sin{\frac{9\pi}{4}} \)
\( 9\left(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \right) \)