Powers and roots of complex numbers

2010013403

Level:

All the solutions of the equation

x^{6} + 3 \sqrt{5} - 6 i = 0

can be displayed as points in the rectangular coordinate system. What is the distance of the two most distant points?

2 \sqrt[3]{3}

2 \sqrt{3}

\sqrt{3}

\sqrt[3]{9}

2 \sqrt[3]{9}

2010013402

Level:

Solve the following equation in the set of complex numbers. (Solve the equation by substitution.)

(3 x + 2)^{4} - 81 = 0

{- \frac{5}{3}; \frac{1}{3}; - \frac{2}{3} + i; - \frac{2}{3} - i}

{- \frac{5}{3}; \frac{1}{3}; \frac{2}{3} + i; \frac{2}{3} - i}

{\frac{5}{3}; - \frac{1}{3}; - \frac{2}{3} + i; - \frac{2}{3} - i}

{\frac{5}{3}; - \frac{1}{3}; \frac{2}{3} + i; \frac{2}{3} - i}

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

2010004617

Level:

Let

z \in C

. The value of the argument of

z^{5}

300^{\circ}

and

| z |^{5} = \frac{1}{32}

. Find

z

z = \frac{1}{4} (1 + i \sqrt{3})

z = \frac{1}{4} (1 - i \sqrt{3})

z = - \frac{1}{2} i

z = \frac{1}{2} (\cos 60^{\circ} - i \sin 60^{\circ})

2010004616

Level:

Let

z \in C

. The value of the argument of

z^{6}

270^{\circ}

and

| z |^{6} = 27

. Find

z

z = \frac{\sqrt{6}}{2} (1 + i)

z = \frac{\sqrt{6}}{2} (1 - i)

z = \sqrt{3} i

z = 3 (\cos 45^{\circ} + i \sin 45^{\circ})

2010004615

Level:

Evaluate

(- 2 + 2 i)^{8}

and write the result in algebraic form.

2^{12}

2^{12} i

- 2^{12} i

- 2^{12}

2010004614

Level:

Evaluate

(1 - i)^{100}

and write the result in algebraic form.

- 2^{50}

2^{50}

- 2^{50} i

2^{50} i

2010004613

Level:

Find the absolute value and the value of the argument of

z = (2 + i \sqrt{12})^{5}

| z | = 1024

;

φ = \frac{5}{3} π

| z | = 512

;

φ = \frac{5}{3} π

| z | = 1024

;

φ = \frac{π}{3}

| z | = 4

;

φ = \frac{5}{3} π

2010004612

Level:

Find the absolute value and the value of the argument of

z = (1 - i \sqrt{3})^{4}

| z | = 16

;

φ = \frac{2}{3} π

| z | = 8

;

φ = \frac{π}{3}

| z | = 256

;

φ = \frac{2}{3} π

| z | = 2

;

φ = \frac{2}{3} π

2010004611

Level:

Identify the complex number that equals to

(- 1 + i \sqrt{3})^{67}

2^{66} (- 1 + i \sqrt{3})

2^{66} (1 - i \sqrt{3})

2^{66} (\sqrt{3} + i)

2^{66} (\sqrt{3} - i)

Powers and roots of complex numbers

2010013403

2010013402

2010013401

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2010004616

2010004615

2010004614

2010004613

2010004612

2010004611

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