9000070108
Level:
A
Evaluate the following complex number.
\[
\left (\frac{1}
{2} + \frac{\sqrt{3}}
{2} \mathrm{i}\right )^{6}
\]
\(1\)
\(- 1\)
\(\mathrm{i}\)
\(-\mathrm{i}\)
Powers and Roots of Complex Numbers
9000070109
Level:
A
Evaluate the following complex number.
\[
\left (\sqrt{3} -\mathrm{i}\right )^{3}
\]
\(- 8\mathrm{i}\)
\(8\)
\(- 8\)
\(8\mathrm{i}\)
9000037404
Level:
A
Given \(z = \sqrt{2}\left (\cos \frac{\pi }{3} -\mathrm{i}\sin \frac{\pi }{3}\right )\),
find \(z^{2}\).
\(- 1 -\mathrm{i}\sqrt{3}\)
\(1 + \mathrm{i}\sqrt{3}\)
\(- 2 -\mathrm{i}\sqrt{2}\)
\(2 + \mathrm{i}\sqrt{2}\)
9000037403
Level:
A
Given \(z = \sqrt{3}\left (\cos \frac{\pi }{3} + \mathrm{i}\sin \frac{\pi }{3}\right )\),
find \(z^{4}\).
\(-\frac{9}
{2} -\frac{9\mathrm{i}\sqrt{3}}
{2} \)
\(\frac{9}
{2} + \frac{9\mathrm{i}\sqrt{3}}
{2} \)
\(\frac{3}
{2}\)
\(-\frac{3}
{2}\)
9000037405
Level:
A
Evaluate \(\left (1 + \mathrm{i}\right )^{7}\).
\(8 - 8\mathrm{i}\)
\(7 - 7\mathrm{i}\)
\(1\)
\(-\mathrm{i}\)
9000037406
Level:
A
Evaluate \(\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )^{40}\).
\(1\)
\(1 + \mathrm{i}\)
\(\mathrm{i}\)
\(1 -\mathrm{i}\)
9000037407
Level:
A
Evaluate \(\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )^{13}\) and find the algebraic form of the result.
\(\mathrm{i}\)
\(1 + 2\mathrm{i}\)
\(1 -\mathrm{i}\)
\(1\)
9000037410
Level:
A
Evaluate \(\left (1 -\mathrm{i}\right )^{3}\).
\(- 2 - 2\mathrm{i}\)
\(2 + 2\mathrm{i}\)
\(1 + \mathrm{i}\)
\(\mathrm{i}\)
9000035709
Level:
A
Simplify \((1 -\mathrm{i})^{-3}\).
\(-\frac{1}
{4} + \frac{1}
{4}\mathrm{i}\)
\(1 + 3\mathrm{i}\)
\(- 2 - 2\mathrm{i}\)
\(\frac{1}
{2} + \frac{1}
{2}\mathrm{i}\)
9000035809
Level:
A
Given the complex number \(z = -1 + \mathrm{i}\),
find the angle in the polar form of the number
\(z^{6}\).
\(\frac{\pi }
{2}\)
\(\frac{3\pi }
{2}\)
\(\frac{3\pi }
{4}\)
\(\frac{7\pi }
{4}\)