Powers and roots of complex numbers

2010013413

Level:

Which of the given numbers does not belong to the solution set of the following equation?

x^{4} + 1 + i = 0

\sqrt[8]{2} (\cos \frac{3 π}{16} + i \sin \frac{3 π}{16})

i \sqrt[4]{- 1 - i}

\sqrt[4]{- 1 - i}

\sqrt[8]{2} (\cos \frac{5 π}{16} + i \sin \frac{5 π}{16})

2010013412

Level:

Which of the given numbers does not belong to the solution set of the following equation?

x^{4} - 1 + i = 0

\sqrt[8]{2} (\cos \frac{3 π}{16} + i \sin \frac{3 π}{16})

- \sqrt[4]{1 - i}

- i \sqrt[4]{1 - i}

\sqrt[8]{2} (\cos (- \frac{π}{16}) + i \sin (- \frac{π}{16}))

2010013411

Level:

Consider the equation

x^{4} = (2 + i)^{4} .

Find the product of all its roots in the set of complex numbers.

7 - 24 i

- 7 + 24 i

16

- 16

2010013410

Level:

Consider the equation

x^{4} = (2 - i)^{4} .

Find the product of all its roots in the set of complex numbers.

7 + 24 i

- 7 - 24 i

16

- 16

2010013409

Level:

Three solutions of the equation

x^{4} + 8 i = 0

are

\begin{array}{r} x_{1} = \sqrt[4]{8} (\cos \frac{3}{8} π + i \sin \frac{3}{8} π), \\ x_{2} = \sqrt[4]{8} (\cos \frac{7}{8} π + i \sin \frac{7}{8} π), \\ x_{3} = \sqrt[4]{8} (\cos \frac{15}{8} π + i \sin \frac{15}{8} π) . \end{array}

Find the fourth solution.

x_{4} = \sqrt[4]{8} (\cos \frac{11}{8} π + i \sin \frac{11}{8} π)

x_{4} = \sqrt[4]{8} (\cos \frac{9}{8} π + i \sin \frac{9}{8} π)

x_{4} = \sqrt[4]{8} (\cos \frac{5}{8} π + i \sin \frac{5}{8} π)

x_{4} = \sqrt[4]{8} (\cos \frac{1}{8} π + i \sin \frac{1}{8} π)

2010013408

Level:

Three solutions of the equation

x^{4} - 2 i = 0

are

\begin{array}{r} x_{1} = \sqrt[4]{2} (\cos \frac{1}{8} π + i \sin \frac{1}{8} π), \\ x_{2} = \sqrt[4]{2} (\cos \frac{5}{8} π + i \sin \frac{5}{8} π), \\ x_{3} = \sqrt[4]{2} (\cos \frac{9}{8} π + i \sin \frac{9}{8} π) . \end{array}

Find the fourth solution.

x_{4} = \sqrt[4]{2} (\cos \frac{13}{8} π + i \sin \frac{13}{8} π)

x_{4} = \sqrt[4]{2} (\cos \frac{11}{8} π + i \sin \frac{11}{8} π)

x_{4} = \sqrt[4]{2} (\cos \frac{15}{8} π + i \sin \frac{15}{8} π)

x_{4} = \sqrt[4]{2} (\cos \frac{3}{8} π + i \sin \frac{3}{8} π)

2010013407

Level:

Two solutions of the equation

x^{3} + 1 - i = 0

are

\begin{aligned} x_{1} & = \sqrt[6]{2} (\cos \frac{π}{4} + i \sin \frac{π}{4}), \\ x_{2} & = \sqrt[6]{2} (\cos \frac{11}{12} π + i \sin \frac{11}{12} π) . \end{aligned}

Find the third solution.

x_{3} = \sqrt[6]{2} (\cos \frac{19}{12} π + i \sin \frac{19}{12} π)

x_{3} = \sqrt[6]{2} (\cos \frac{7}{12} π + i \sin \frac{7}{12} π)

x_{3} = \sqrt[6]{2} (\cos \frac{5}{12} π + i \sin \frac{5}{12} π)

x_{3} = \sqrt[6]{2} (\cos \frac{13}{12} π + i \sin \frac{13}{12} π)

2010013406

Level:

Find the solution set of the following equation in the set of complex numbers.

x^{3} + 8 i = 0

{2 i; \sqrt{3} - i; - \sqrt{3} - i}

{- 2 i; \sqrt{3} - i; - \sqrt{3} - i}

{- 2; - \sqrt{3} + i; \sqrt{3} + i}

{2; - \sqrt{3} + i; \sqrt{3} + i}

2010013405

Level:

Find the solution set of the following equation in the set of complex numbers.

x^{3} + 27 = 0

{- 3; \frac{3}{2} - i \frac{3 \sqrt{3}}{2}; \frac{3}{2} + i \frac{3 \sqrt{3}}{2}}

{- 3; - \frac{3}{2} + i \frac{3 \sqrt{3}}{2}; - \frac{3}{2} - i \frac{3 \sqrt{3}}{2}}

{- 3; \frac{3}{2} - i \frac{\sqrt{3}}{2}; \frac{3}{2} + i \frac{\sqrt{3}}{2}}

{- 3; - \frac{3}{2} + i \frac{\sqrt{3}}{2}; - \frac{3}{2} - i \frac{\sqrt{3}}{2}}

2010013404

Level:

Find the solution set of the following equation in the set of complex numbers.

x^{3} - 8 = 0

{2; - 1 - i \sqrt{3}; - 1 + i \sqrt{3}}

{2; 1 - i \sqrt{3}; 1 + i \sqrt{3}}

{2; 1 - i \sqrt{3}}

{2; - 1 + i \sqrt{3}}

Powers and roots of complex numbers

2010013413

2010013412

2010013411

2010013410

2010013409

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2010013407

2010013406

2010013405

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