Powers and roots of complex numbers

1103109303

Level:

Consider an equation

x^{n} + b = 0

, where

n

is a positive integer and

b

is a real number. The points that correspond to the roots of the equation are marked in the figure as black points. Find the equation.

x^{8} - 256 = 0

x^{8} + 256 = 0

x^{4} + 16 = 0

x^{4} - 16 = 0

x^{6} - 64 = 0

x^{6} + 64 = 0

1103109304

Level:

Choose the figure in which the points marked in black correspond to the roots of the equation

x^{5} + 1 = 0

2000002601

Level:

Consider the equation

x^{4} + 16 = 0

. Which of the given numbers is the solution of this equation?

2 (\cos \frac{π}{4} + i \sin \frac{π}{4})

2 i

- 2 i

2 (\cos π + i \sin π)

2000002602

Level:

Consider the equation

x^{4} = 1

, where

x

is a complex variable. Which of the following statements is true?

The equation has four different complex roots.

The equation has no real root.

The equation has two double roots:

x_{1, 2} = 1

and

x_{3, 4} = - 1

The equation has the root

x = 1 + i

2000002603

Level:

One of the roots of the equation

x^{3} - 8 = 0

x_{1} = - 1 - i \sqrt{3}

. Find the sum of all its roots.

0

- 8

- 2 i \sqrt{3}

- 4

2000002604

Level:

Find the solution set of the equation

x^{4} + 81 = 0

if you know that one of its roots is

\frac{3}{\sqrt{2}} (1 + i)

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

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

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

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

2000002605

Level:

How many solutions does the equation

2 x^{4} = 32

have in the set of complex numbers?

four

one

two

eight

2000002606

Level:

Imagine all the solutions of the equation

x^{6} - 64 = 0

shown as points in the complex plane. Find the false statement.

Two points lie on the imaginary axis.

The values of the arguments of any two solutions differ by an integer multiple of

\frac{π}{3}

All solutions of the equation lie on a circle centered at the origin with a radius of

2

Two points lie on the real axis.

2000002608

Level:

Find the right formula for solving the equation

x^{5} + 32 = 0

x_{k} = \sqrt[5]{| - 32 |} (\cos \frac{π + 2 k π}{5} + i \sin \frac{π + 2 k π}{5})

k = 0, 1, 2, 3, 4

x_{k} = \sqrt[5]{- 32} (\cos \frac{π + 2 k π}{5} + i \sin \frac{π + 2 k π}{5})

k = 0, 1, 2, 3, 4

x_{k} = \sqrt[5]{| - 32 |} (\cos \frac{π + k π}{5} + i \sin \frac{π + k π}{5})

k = 0, 1, 2, 3, 4

x_{k} = \sqrt[5]{| - 32 |} (\cos \frac{π + 2 k π}{5} + \sin \frac{π + 2 k π}{5})

k = 0, 1, 2, 3, 4

2010013401

Level:

Consider the equation

x^{4} + 16 = 0

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

0

2

- 2

16

- 16

Powers and roots of complex numbers

1103109303

1103109304

2000002601

2000002602

2000002603

2000002604

2000002605

2000002606

2000002608

2010013401

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