Powers and Roots of Complex Numbers

2010013402

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

2010013404

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

2010013405

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

9000034301

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

9000034302

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

9000034304

Level: 
B
Find the solution set of the following equation in the set of complex numbers. \[ x^{4} - 1 = 0 \]
\(\{1,\ -1,\ \mathrm{i},\ -\mathrm{i}\}\)
\(\{1 -\mathrm{i},\ -1 -\mathrm{i}\}\)
\(\{1 + \mathrm{i},\ -1 + \mathrm{i}\}\)
\(\{1 + \mathrm{i},\ 1 -\mathrm{i},\ -1 + \mathrm{i},\ -1 -\mathrm{i}\}\)

9000034305

Level: 
B
Solve the following equation in the set of complex numbers. \[ x^{4} + 16 = 0 \]
\(x_{1, 2} = \sqrt{2}(1\pm \mathrm{i}),\ x_{3, 4} = -\sqrt{2}(1\pm \mathrm{i})\)
\(x_{1, 2} = 1\pm \mathrm{i},\ x_{3, 4} = -1\pm \mathrm{i}\)
\(x_{1, 2} = 2(1\pm \mathrm{i}),\ x_{3, 4} = -2(1\pm \mathrm{i})\)
\(x_{1, 2} = \frac{\sqrt{2}} {2} (1\pm \mathrm{i}),\ x_{3, 4} = -\frac{\sqrt{2}} {2} (1\pm \mathrm{i})\)

9000034306

Level: 
B
Solve the following equation in the set of complex numbers. \[ x^{6} - 64 = 0 \]
\(x_{1, 2} =\pm 2,\ x_{3, 4} = 1\pm \mathrm{i}\sqrt{3},\ x_{5, 6} = -1\pm \mathrm{i}\sqrt{3}\)
\(x_{1, 2} =\pm 2,\ x_{3, 4} = \frac{1} {2}\pm \mathrm{i}\frac{\sqrt{3}} {2} ,\ x_{5, 6} = -\frac{1} {2}\pm \mathrm{i}\frac{\sqrt{3}} {2} \)
\(x_{1, 2} =\pm 4,\ x_{3, 4} = 1\pm \mathrm{i}\sqrt{3},\ x_{5, 6} = -1\pm \mathrm{i}\sqrt{3}\)
\(x_{1, 2} =\pm 8,\ x_{3, 4} = 2\pm 2\mathrm{i}\sqrt{3},\ x_{5, 6} = -2\pm 2\mathrm{i}\sqrt{3}\)

1003118401

Level: 
C
Find the solution set of the equation \( x^3 - 8\mathrm{i} = 0 \) in the set of complex numbers.
\( \left\{\sqrt3+\mathrm{i}, -\sqrt3+\mathrm{i},-2\mathrm{i} \right\} \)
\( \left\{ 2\mathrm{i}, -\sqrt3-\mathrm{i}, \sqrt3-\mathrm{i} \right\} \)
\( \left\{\frac{\sqrt3}2+\frac12\mathrm{i}, -\frac{\sqrt3}2+\frac12\mathrm{i},-\mathrm{i} \right\} \)
\( \left\{\mathrm{i},-\frac{\sqrt3}2-\frac12\mathrm{i}, \frac{\sqrt3}2-\frac12\mathrm{i} \right\} \)

1003118402

Level: 
C
Which of the given complex numbers is not a root of the equation \( x^6 + 8\mathrm{i} = 0 \)?
\( 1-\mathrm{i} \)
\( 1+\mathrm{i} \)
\( -1-\mathrm{i} \)
\( \sqrt2\left(\cos\frac{7\pi}{12}+\mathrm{i}\cdot\sin\frac{7\pi}{12}\right) \)
\( \sqrt2\left(\cos\frac{23\pi}{12}+\mathrm{i}\cdot\sin\frac{23\pi}{12}\right) \)