Powers and roots of complex numbers

9000035808

Level:

Evaluate

(1 - i)^{10}

- 32 i

32

32 i

- 32

9000035810

Level:

Given the complex number

z = - 2 + 2 i

, find all the roots of

\sqrt[3]{z}

\begin{aligned} w_{0} = \sqrt[6]{8} (\cos \frac{π}{4} + i \sin \frac{π}{4}) \\ w_{1} = \sqrt[6]{8} (\cos \frac{11 π}{12} + i \sin \frac{11 π}{12}) \\ w_{2} = \sqrt[6]{8} (\cos \frac{19 π}{12} + i \sin \frac{19 π}{12}) \end{aligned}

\begin{aligned} w_{0} = 2 (\cos \frac{π}{4} + i \sin \frac{π}{4}) \\ w_{1} = 2 (\cos \frac{11 π}{12} + i \sin \frac{11 π}{12}) \\ w_{2} = 2 (\cos \frac{19 π}{12} + i \sin \frac{19 π}{12}) \end{aligned}

\sqrt[3]{- 2} + \sqrt[3]{2}

\begin{aligned} w_{0} = 2 (\cos \frac{π}{3} + i \sin \frac{π}{3}) \\ w_{1} = 2 (\cos π + i \sin π) \\ w_{2} = 2 (\cos \frac{5 π}{3} + i \sin \frac{5 π}{3}) \end{aligned}

9000034302

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

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

9000034303

Level:

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

x^{3} + i = 0

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

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

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

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

9000034304

Level:

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

x^{4} - 1 = 0

{1; - 1; i; - i}

{1 - i; - 1 - i}

{1 + i; - 1 + i}

{1 + i; 1 - i; - 1 + i; - 1 - i}

9000034305

Level:

Solve the following equation in the set of complex numbers.

x^{4} + 16 = 0

x_{1, 2} = \sqrt{2} (1 \pm i), x_{3, 4} = - \sqrt{2} (1 \pm i)

x_{1, 2} = 1 \pm i, x_{3, 4} = - 1 \pm i

x_{1, 2} = 2 (1 \pm i), x_{3, 4} = - 2 (1 \pm i)

x_{1, 2} = \frac{\sqrt{2}}{2} (1 \pm i), x_{3, 4} = - \frac{\sqrt{2}}{2} (1 \pm i)

9000034306

Level:

Solve the following equation in the set of complex numbers.

x^{6} - 64 = 0

x_{1, 2} = \pm 2, x_{3, 4} = 1 \pm i \sqrt{3}, x_{5, 6} = - 1 \pm i \sqrt{3}

x_{1, 2} = \pm 2, x_{3, 4} = \frac{1}{2} \pm i \frac{\sqrt{3}}{2}, x_{5, 6} = - \frac{1}{2} \pm i \frac{\sqrt{3}}{2}

x_{1, 2} = \pm 4, x_{3, 4} = 1 \pm i \sqrt{3}, x_{5, 6} = - 1 \pm i \sqrt{3}

x_{1, 2} = \pm 8, x_{3, 4} = 2 \pm 2 i \sqrt{3}, x_{5, 6} = - 2 \pm 2 i \sqrt{3}

9000034307

Level:

Let

x

be any of the complex solutions of the following equation.

x^{5} + \sqrt{3} - i = 0

Find the absolute value of

x

\sqrt[5]{2}

2

\sqrt[5]{4}

\sqrt[10]{2}

9000034308

Level:

Two solutions of the equation

x^{3} + 1 + i = 0

are

\begin{aligned} x_{1} & = \sqrt[6]{2} (\cos \frac{5}{12} π + i \sin \frac{5}{12} π), \\ x_{2} & = \sqrt[6]{2} (\cos \frac{13}{12} π + i \sin \frac{13}{12} π) . \end{aligned}

Find the third solution.

x_{3} = \sqrt[6]{2} (\cos \frac{21}{12} π + i \sin \frac{21}{12} π)

x_{3} = \sqrt[6]{2} (\cos \frac{9}{12} π + i \sin \frac{9}{12} π)

x_{3} = \sqrt[6]{2} (\cos \frac{17}{12} π + i \sin \frac{17}{12} π)

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

9000034309

Level:

Find the angle

φ

such that the angles in the polar form of any two solutions of the equation

x^{5} - 1 + i \sqrt{3} = 0

differ by an integer multiple of

φ

φ = \frac{2}{5} π

φ = \frac{3}{5} π

φ = \frac{4}{5} π

φ = π

Powers and roots of complex numbers

9000035808

9000035810

9000034302

9000034303

9000034304

9000034305

9000034306

9000034307

9000034308

9000034309

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