Powers and roots of complex numbers

1003123401

Level: 
A
Given the complex number \( a =\sqrt3\cdot\left( \cos 225^{\circ} + \mathrm{i}\cdot\sin 225^{\circ}\right) \), find the polar form of \( a^6 \).
\( 27\cdot\left(\cos270^{\circ}+\mathrm{i}\cdot\sin270^{\circ}\right) \)
\( 9\cdot\left(\cos90^{\circ}+\mathrm{i}\cdot\sin90^{\circ}\right) \)
\( 27\cdot\left(\cos90^{\circ}+\mathrm{i}\cdot\sin90^{\circ}\right) \)
\( 9\cdot\left(\cos270^{\circ}+\mathrm{i}\cdot\sin270^{\circ}\right) \)

1003123402

Level: 
A
Given the complex number \( b=\sqrt[3]2\cdot\left(\cos\frac56\pi+\mathrm{i}\cdot\sin\frac56\pi\right) \), find the polar form of \( b^9 \).
\( 8\cdot\left(\cos\frac32\pi+\mathrm{i}\cdot\sin\frac32\pi\right) \)
\( 64\cdot\left(\cos\frac12\pi-\mathrm{i}\cdot\sin\frac12\pi\right) \)
\( 8\cdot\left(\cos\frac12\pi-\mathrm{i}\cdot\sin\frac12\pi\right) \)
\( 64\cdot\left(\cos\frac32\pi+\mathrm{i}\cdot\sin\frac32\pi\right) \)

2000002102

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

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 \(\varphi \in (-\pi; \pi] \).
\( -\frac{\pi}{2} \)
\( \frac{\pi}{2} \)
\( 0 \)
\( \frac{3}{26}\pi \)