Powers and roots of complex numbers

2010013413

Level: 
C
Which of the given numbers does not belong to the solution set of the following equation? \[x^{4}+1+\mathrm{i}=0\]
\(\root{8}\of{2}\left (\cos \frac{3\pi}{16} + \mathrm{i}\sin \frac{3\pi}{16}\right )\)
\(\mathrm{i}\root{4}\of{-1-\mathrm{i}}\)
\(\root{4}\of{-1-\mathrm{i}}\)
\(\root{8}\of{2}\left (\cos \frac{5\pi}{16} + \mathrm{i}\sin \frac{5\pi}{16}\right )\)

2010013412

Level: 
C
Which of the given numbers does not belong to the solution set of the following equation? \[x^{4}-1+\mathrm{i}=0\]
\(\root{8}\of{2}\left (\cos \frac{3\pi}{16} + \mathrm{i}\sin \frac{3\pi}{16}\right )\)
\(-\root{4}\of{1-\mathrm{i}}\)
\(-\mathrm{i}\root{4}\of{1-\mathrm{i}}\)
\(\root{8}\of{2}\left (\cos \left(-\frac{\pi}{16}\right) + \mathrm{i}\sin \left(-\frac{\pi}{16}\right)\right )\)

2010013409

Level: 
C
Three solutions of the equation \[ x^{4} + 8\mathrm{i} = 0 \] are \[\begin{aligned}x_{1} = \root{4}\of{8}\left (\cos \frac{3}{8}\pi + \mathrm{i}\sin \frac{3}{8}\pi \right ), \\ x_{2} = \root{4}\of{8}\left (\cos \frac{7}{8}\pi + \mathrm{i}\sin \frac{7}{8}\pi \right ),\\ x_{3} = \root{4}\of{8}\left (\cos \frac{15}{8}\pi + \mathrm{i}\sin \frac{15}{8}\pi \right ).\\ \end{aligned}\] Find the fourth solution.
\(x_{4} = \root{4}\of{8}\left (\cos \frac{11}{8}\pi + \mathrm{i}\sin \frac{11}{8}\pi \right )\)
\(x_{4} = \root{4}\of{8}\left (\cos \frac{9}{8}\pi + \mathrm{i}\sin \frac{9}{8}\pi \right )\)
\(x_{4} = \root{4}\of{8}\left (\cos \frac{5}{8}\pi + \mathrm{i}\sin \frac{5}{8}\pi \right )\)
\(x_{4} = \root{4}\of{8}\left (\cos \frac{1}{8}\pi + \mathrm{i}\sin \frac{1}{8}\pi \right )\)

2010013408

Level: 
C
Three solutions of the equation \[ x^{4} - 2\mathrm{i} = 0 \] are \[\begin{aligned}x_{1} = \root{4}\of{2}\left (\cos \frac{1}{8}\pi + \mathrm{i}\sin \frac{1}{8}\pi \right ),\\ x_{2} = \root{4}\of{2}\left (\cos \frac{5}{8}\pi + \mathrm{i}\sin \frac{5}{8}\pi \right ),\\ x_{3} = \root{4}\of{2}\left (\cos \frac{9}{8}\pi + \mathrm{i}\sin \frac{9}{8}\pi \right ).\\ \end{aligned}\] Find the fourth solution.
\(x_{4} = \root{4}\of{2}\left (\cos \frac{13}{8}\pi + \mathrm{i}\sin \frac{13}{8}\pi \right )\)
\(x_{4} = \root{4}\of{2}\left (\cos \frac{11}{8}\pi + \mathrm{i}\sin \frac{11}{8}\pi \right )\)
\(x_{4} = \root{4}\of{2}\left (\cos \frac{15}{8}\pi + \mathrm{i}\sin \frac{15}{8}\pi \right )\)
\(x_{4} = \root{4}\of{2}\left (\cos \frac{3}{8}\pi + \mathrm{i}\sin \frac{3}{8}\pi \right )\)

2010013407

Level: 
C
Two solutions of the equation \[ x^{3} + 1 - \mathrm{i} = 0 \] are \[ \begin{aligned}x_{1}& = \root{6}\of{2}\left (\cos \frac{\pi} {4} + \mathrm{i}\sin \frac{\pi} {4} \right ),& \\x_{2}& = \root{6}\of{2}\left (\cos \frac{11} {12}\pi + \mathrm{i}\sin \frac{11} {12}\pi \right ). \\ \end{aligned} \] Find the third solution.
\(x_{3} = \root{6}\of{2}\left (\cos \frac{19} {12}\pi + \mathrm{i}\sin \frac{19} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{7} {12}\pi + \mathrm{i}\sin \frac{7} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{5} {12}\pi + \mathrm{i}\sin \frac{5} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{13} {12}\pi + \mathrm{i}\sin \frac{13} {12}\pi \right )\)

2010013406

Level: 
C
Find the solution set of the following equation in the set of complex numbers. \[ x^{3} + 8\mathrm{i} = 0 \]
\(\left\{2\mathrm{i};\ \sqrt{3} -\mathrm{i};\ -\sqrt{3}-\mathrm{i}\right\}\)
\(\left\{ -2\mathrm{i};\ \sqrt{3} -\mathrm{i};\ -\sqrt{3}-\mathrm{i}\right\}\)
\(\left\{ -2;\ -\sqrt{3} +\mathrm{i};\ \sqrt{3}+\mathrm{i}\right\}\)
\(\left\{ 2;\ -\sqrt{3} +\mathrm{i};\ \sqrt{3}+\mathrm{i}\right\}\)

2010013405

Level: 
B
Find the solution set of the following equation in the set of complex numbers. \[ x^{3} + 27 = 0 \]
\(\left\{-3;\ \frac32 - \mathrm{i}\frac{3\sqrt3} {2} ;\ \frac32 +\mathrm{i}\frac{3\sqrt{3}} {2} \right\}\)
\(\left\{-3;\ -\frac32 + \mathrm{i}\frac{3\sqrt3} {2} ;\ -\frac32 -\mathrm{i}\frac{3\sqrt3} {2} \right\}\)
\(\left\{-3;\ \frac32 - \mathrm{i}\frac{\sqrt3} {2} ;\ \frac32 +\mathrm{i}\frac{\sqrt3} {2} \right\}\)
\(\left\{-3;\ -\frac32 + \mathrm{i}\frac{\sqrt3} {2} ;\ -\frac32 -\mathrm{i}\frac{\sqrt3} {2} \right\}\)

2010013404

Level: 
B
Find the solution set of the following equation in the set of complex numbers. \[ x^{3} - 8 = 0 \]
\(\left\{2;\ -1 - \mathrm{i}\sqrt{3};\ -1 +\mathrm{i}\sqrt{3}\right\}\)
\(\left\{2;\ 1 - \mathrm{i}\sqrt{3};\ 1 +\mathrm{i}\sqrt{3}\right\}\)
\(\left\{2;\ 1 - \mathrm{i}\sqrt{3}\right\}\)
\(\left\{2;\ - 1 + \mathrm{i}\sqrt{3}\right\}\)