Powers and roots of complex numbers

2000002109

Level:

Given

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

, determine which of the numbers below does not represent

z^{6}

9

9 i

9 (\cos \frac{9 π}{2} + i \sin \frac{9 π}{2})

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

2000002110

Level:

Find the algebraic form of the complex number

{(\cos \frac{π}{6} + i \sin \frac{π}{6})}^{- 6}

- 1

i

- i

1

2010004601

Level:

Evaluate

{(\cos \frac{π}{4} + i \sin \frac{π}{4})}^{40}

and find the algebraic form of the result.

1

- 1

- i

i

2010004602

Level:

Identify a complex number that is not equal to

{(\cos \frac{π}{3} + i \sin \frac{π}{3})}^{19}

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

\cos \frac{π}{3} + i \sin \frac{π}{3}

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

\cos (- \frac{5 π}{3}) + i \sin (- \frac{5 π}{3})

2010004603

Level:

Identify a complex number that is not equal to

{(\cos \frac{π}{6} + i \sin \frac{π}{6})}^{13}

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

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

\cos \frac{π}{6} + i \sin \frac{π}{6}

\cos \frac{25 π}{6} + i \sin \frac{25 π}{6}

2010004604

Level:

Find the algebraic form of the complex number:

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

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

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

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

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

2010004605

Level:

Find the algebraic form of the complex number:

{(\frac{1}{2} + \sin \frac{π}{6} + i \sin \frac{π}{2})}^{3}

- 2 + 2 i

2 - 2 i

- 2 - 2 i

2 + 2 i

2010004606

Level:

Evaluate

(\cos 33^{\circ} + i \sin 33^{\circ})^{10}

and find the algebraic form of the result.

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

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

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

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

2010004607

Level:

Evaluate

(\cos 27^{\circ} + i \sin 27^{\circ})^{5}

and find the algebraic form of the result.

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

2010004608

Level:

Evaluate

(1 + i)^{22}

and write the result in algebraic form.

- 2^{11} i

2^{11} i

2^{22} i

- 2^{22} i

Powers and roots of complex numbers

2000002109

2000002110

2010004601

2010004602

2010004603

2010004604

2010004605

2010004606

2010004607

2010004608

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