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 \( z^6\).
\( 9 \)
\( 9i \)
\( 9\left(\cos{\frac{9\pi}{2}}+i\sin{\frac{9\pi}{2}}\right) \)
\( 9\left(\cos{\frac{\pi}{2}}+i\sin{\frac{\pi}{2}}\right) \)

2010004602

Level: 
A
Identify a complex number that is not equal to \(\left (\cos \frac{\pi }{3} + \mathrm{i}\sin \frac{\pi }{3}\right )^{19}\).
\( \frac{1}{2} - \frac{\sqrt{3}}{2}\mathrm{i}\)
\( \cos \frac{\pi }{3} + \mathrm{i}\sin \frac{\pi }{3} \)
\( \frac{1}{2} + \frac{\sqrt{3}}{2}\mathrm{i}\)
\( \cos \left(-\frac{5\pi }{3}\right) + \mathrm{i}\sin \left(-\frac{5\pi }{3}\right) \)

2010004603

Level: 
A
Identify a complex number that is not equal to \(\left (\cos \frac{\pi }{6} + \mathrm{i}\sin \frac{\pi }{6}\right )^{13}\).
\( \frac{\sqrt{3}}{2} - \frac{1}{2}\mathrm{i}\)
\( \frac{\sqrt{3}}{2} + \frac{1}{2}\mathrm{i}\)
\( \cos \frac{\pi }{6} + \mathrm{i}\sin \frac{\pi }{6} \)
\( \cos \frac{25\pi }{6} + \mathrm{i}\sin \frac{25\pi }{6} \)

2010004604

Level: 
A
Find the algebraic form of the complex number: \[ \left (\cos \frac{\pi } {3} + \mathrm{i}\sin \frac{\pi } {3}\right )^{5} \]
\(\frac{1} {2} - \mathrm{i}\frac{\sqrt{3}} {2} \)
\(\frac{\sqrt{3}} {2} +\mathrm{i} \frac{1} {2} \)
\(\frac{\sqrt{3}} {2} -\mathrm{i} \frac{1} {2} \)
\(\frac{1} {2} + \mathrm{i}\frac{\sqrt{3}} {2} \)

2010004606

Level: 
A
Evaluate \( (\cos 33^{\circ} + \mathrm{i}\sin 33^{\circ})^{10} \) and find the algebraic form of the result.
\( \frac{\sqrt{3}}{2} - \frac{1}{2}\mathrm{i}\)
\( \frac{\sqrt{3}}{2} + \frac{1}{2}\mathrm{i}\)
\( \frac{1}{2} + \frac{\sqrt{3}}{2}\mathrm{i}\)
\( \frac{1}{2} - \frac{\sqrt{3}}{2}\mathrm{i}\)

2010004607

Level: 
A
Evaluate \( (\cos 27^{\circ} + \mathrm{i}\sin 27^{\circ})^{5} \) and find the algebraic form of the result.
\( -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}\mathrm{i}\)
\( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}\mathrm{i}\)
\( -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}\mathrm{i}\)
\( \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}\mathrm{i}\)