Powers and roots of complex numbers

2010013402

Level: 
B
Solve the following equation in the set of complex numbers. (Solve the equation by substitution.) \[ (3x + 2)^4 - 81 = 0 \]
\( \left\{-\frac53;\frac13;-\frac23+\mathrm{i} ;-\frac23-\mathrm{i} \right\} \)
\( \left\{-\frac53;\frac13;\frac23+\mathrm{i} ;\frac23-\mathrm{i} \right\} \)
\( \left\{\frac53;-\frac13;-\frac23+\mathrm{i} ;-\frac23-\mathrm{i} \right\} \)
\( \left\{\frac53;-\frac13;\frac23+\mathrm{i} ;\frac23-\mathrm{i} \right\} \)

2010004617

Level: 
A
Let \( z \in \mathbb{C}\). The value of the argument of \(z^5\) is \(300^{\circ}\) and \(|z|^5=\frac1{32}\). Find \(z\).
\( z=\frac{1}{4}(1+\mathrm{i}\sqrt{3})\)
\( z=\frac{1}{4}(1-\mathrm{i}\sqrt{3})\)
\( z=-\frac{1}{2}\mathrm{i}\)
\( z=\frac{1}{2}(\cos 60^{\circ} - \mathrm{i} \sin 60^{\circ})\)

2010004616

Level: 
A
Let \( z \in \mathbb{C}\). The value of the argument of \(z^6\) is \(270^{\circ}\) and \(|z|^6=27\). Find \(z\).
\( z=\frac{\sqrt{6}}{2}(1+\mathrm{i})\)
\( z=\frac{\sqrt{6}}{2}(1-\mathrm{i})\)
\( z=\sqrt{3}\mathrm{i}\)
\( z=3(\cos 45^{\circ} + \mathrm{i} \sin 45^{\circ})\)