Powers and roots of complex numbers

1003123401

Level:

Given the complex number

a = \sqrt{3} \cdot (\cos 225^{\circ} + i \cdot \sin 225^{\circ})

, find the polar form of

a^{6}

27 \cdot (\cos 270^{\circ} + i \cdot \sin 270^{\circ})

9 \cdot (\cos 90^{\circ} + i \cdot \sin 90^{\circ})

27 \cdot (\cos 90^{\circ} + i \cdot \sin 90^{\circ})

9 \cdot (\cos 270^{\circ} + i \cdot \sin 270^{\circ})

1003123402

Level:

Given the complex number

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

, find the polar form of

b^{9}

8 \cdot (\cos \frac{3}{2} π + i \cdot \sin \frac{3}{2} π)

64 \cdot (\cos \frac{1}{2} π - i \cdot \sin \frac{1}{2} π)

8 \cdot (\cos \frac{1}{2} π - i \cdot \sin \frac{1}{2} π)

64 \cdot (\cos \frac{3}{2} π + i \cdot \sin \frac{3}{2} π)

2000002101

Level:

Find the algebraic form of the complex number

{((\sqrt[3]{4} (\cos \frac{π}{6} + i \sin \frac{π}{6}))}^{3}

4 i

- 4 i

4

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

2000002102

Level:

Consider

z = \cos \frac{π}{4} + i \sin \frac{π}{4}

and find

z^{9}

\cos \frac{π}{4} + i \sin \frac{π}{4}

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

\cos \frac{9 π}{4} - i \sin \frac{9 π}{4}

9 (\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2})

2000002103

Level:

Find the algebraic form of the complex number

{(\sqrt{2} (\cos \frac{π}{2} + i \sin \frac{π}{2}))}^{18}

- 512

512 i

512

- 512 i

2000002104

Level:

Find the algebraic form of the complex number

{(\cos (- \frac{π}{3}) + i \sin (- \frac{π}{3}))}^{99}

- 1

- i

1

i

2000002105

Level:

Find the value of the principal argument

φ

{(\cos \frac{2 π}{3} + i \sin \frac{2 π}{3})}^{18}

. The principal value is the corresponding angle

φ \in (- π; π]

0

π

- π

\frac{π}{3}

2000002106

Level:

Which of the following numbers is not the value of the argument of

{(\cos \frac{π}{4} + i \sin \frac{π}{4})}^{5}

\frac{π}{4}

\frac{5 π}{4}

- \frac{3 π}{4}

\frac{13 π}{4}

2000002107

Level:

Find the algebraic form of

(\cos 45^{\circ} + i \sin 45^{\circ})^{9}

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

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

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

i + 1

2000002108

Level:

Find the value of the principal argument

φ

{(3 (\cos \frac{3 π}{2} + i \sin \frac{3 π}{2}))}^{13}

. The principal value is the corresponding angle

φ \in (- π; π]

- \frac{π}{2}

\frac{π}{2}

0

\frac{3}{26} π

Powers and roots of complex numbers

1003123401

1003123402

2000002101

2000002102

2000002103

2000002104

2000002105

2000002106

2000002107

2000002108

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