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

1003109302

Level:

Consider the equation

x^{4} - 81 = 0

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

- 81

81

- 3

3

0

1003109301

Level:

Consider the equation

x^{4} + 81 = 0

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

0

3

- 3

81

- 81

9000070102

Level:

Evaluate the following complex number.

{(\cos \frac{π}{3} + i \sin \frac{π}{3})}^{10}

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

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

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

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

9000070103

Level:

Evaluate the following complex number.

{(\cos π + i \sin π)}^{9}

- 1

1

i

- i

9000070104

Level:

Evaluate the following complex number.

{(\sin 2 π + i \cos 2 π)}^{11}

- i

- 1

1

i

9000070105

Level:

Evaluate the following complex number.

i^{13}

\cos \frac{π}{2} + i \sin \frac{π}{2}

\cos \frac{π}{2} - i \sin \frac{π}{2}

\sin \frac{π}{2} + i \cos \frac{π}{2}

\cos \frac{3 π}{2} + i \sin \frac{3 π}{2}

9000070107

Level:

Find the algebraic form of the complex number:

{(\frac{1}{2} + \cos \frac{π}{3} + i \cos 2 π)}^{5}

- 4 - 4 i

- 4 + 4 i

4 - 4 i

4 + 4 i

9000070101

Level:

Find the algebraic form of the complex number:

{(\cos \frac{π}{4} + i \sin \frac{π}{4})}^{3}

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

- \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2}

\frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2}

\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}

9000070106

Level:

Evaluate the following complex number.

(1 - i)^{8}

16

- 16 i

16 i

- 16

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Powers and roots of complex numbers

1103109303

1003109302

1003109301

9000070102

9000070103

9000070104

9000070105

9000070107

9000070101

9000070106

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