Powers and roots of complex numbers

1103109303

Level: 
B
Consider an equation \( x^n+b=0 \), where \( n \) is a positive integer and \( b \) is a real number. The points that correspond to the roots of the equation are marked in the figure as black points. Find the equation.
\( x^8 - 256 = 0 \)
\( x^8 + 256 = 0 \)
\( x^4 + 16 = 0 \)
\( x^4 - 16 = 0 \)
\( x^6 - 64 = 0 \)
\( x^6 + 64 = 0 \)

9000070105

Level: 
A
Evaluate the following complex number. \[ \mathrm{i}^{13} \]
\(\cos \frac{\pi } {2} + \mathrm{i}\sin \frac{\pi } {2}\)
\(\cos \frac{\pi } {2} -\mathrm{i}\sin \frac{\pi } {2}\)
\(\sin \frac{\pi } {2} + \mathrm{i}\cos \frac{\pi } {2}\)
\(\cos \frac{3\pi } {2} + \mathrm{i}\sin \frac{3\pi } {2}\)

9000070101

Level: 
A
Find the algebraic form of the complex number: \[ \left (\cos \frac{\pi } {4} + \mathrm{i}\sin \frac{\pi } {4}\right )^{3} \]
\(-\frac{\sqrt{2}} {2} + \mathrm{i}\frac{\sqrt{2}} {2} \)
\(-\frac{\sqrt{2}} {2} -\mathrm{i}\frac{\sqrt{2}} {2} \)
\(\frac{\sqrt{2}} {2} -\mathrm{i}\frac{\sqrt{2}} {2} \)
\(\frac{\sqrt{2}} {2} + \mathrm{i}\frac{\sqrt{2}} {2} \)