2010004609
Level:
A
Evaluate \( (2+2\mathrm{i})^{100} \) and write the result in algebraic form.
\( -2^{150} \)
\( -2^{150}\mathrm{i} \)
\( 2^{150}\mathrm{i} \)
\( 2^{150}\)
Powers and roots of complex numbers
2010004610
Level:
A
Identify the complex number that equals to \((\sqrt{3} - \mathrm{i})^{37} \).
\(2^{36}(\sqrt{3}-\mathrm{i})\)
\(2^{36}(-1+\mathrm{i}\sqrt{3})\)
\(2^{36}(\sqrt{3}+\mathrm{i})\)
\(2^{36}(-1-\mathrm{i}\sqrt{3})\)
2010004611
Level:
A
Identify the complex number that equals to \(( -1+ \mathrm{i}\sqrt{3})^{67} \).
\(2^{66}(-1+\mathrm{i}\sqrt{3})\)
\(2^{66}(1-\mathrm{i}\sqrt{3})\)
\(2^{66}(\sqrt{3}+\mathrm{i})\)
\(2^{66}(\sqrt{3}-\mathrm{i})\)
2010004612
Level:
A
Find the absolute value and the value of the argument of \( z=(1-\mathrm{i}\sqrt{3})^4\).
\( |z|=16\); \(\varphi = \frac{2}{3}\pi\)
\( |z|=8\); \(\varphi = \frac{\pi}{3}\)
\( |z|=256\); \(\varphi = \frac{2}{3}\pi\)
\( |z|=2\); \(\varphi = \frac{2}{3}\pi\)
2010004613
Level:
A
Find the absolute value and the value of the argument of \( z=(2+\mathrm{i}\sqrt{12})^5\).
\( |z|=1024\); \(\varphi = \frac{5}{3}\pi\)
\( |z|=512\); \(\varphi = \frac{5}{3}\pi\)
\( |z|=1024\); \(\varphi = \frac{\pi}{3}\)
\( |z|=4\); \(\varphi = \frac{5}{3}\pi\)
2010004614
Level:
A
Evaluate \( (1-\mathrm{i})^{100} \) and write the result in algebraic form.
\( -2^{50}\)
\( 2^{50}\)
\( -2^{50}\mathrm{i} \)
\( 2^{50}\mathrm{i} \)
2010004615
Level:
A
Evaluate \( (-2+2\mathrm{i})^{8} \) and write the result in algebraic form.
\( 2^{12}\)
\( 2^{12}\mathrm{i}\)
\( -2^{12}\mathrm{i} \)
\( -2^{12} \)
2010004616
Level:
A
Let \( z \in \mathbb{C}\). The value of the argument of \(z^6\) is \(270^{\circ}\) and \(|z|^6=27\). Find \(z\).
\( z=\frac{\sqrt{6}}{2}(1+\mathrm{i})\)
\( z=\frac{\sqrt{6}}{2}(1-\mathrm{i})\)
\( z=\sqrt{3}\mathrm{i}\)
\( z=3(\cos 45^{\circ} + \mathrm{i} \sin 45^{\circ})\)
2010004617
Level:
A
Let \( z \in \mathbb{C}\). The value of the argument of \(z^5\) is \(300^{\circ}\) and \(|z|^5=\frac1{32}\). Find \(z\).
\( z=\frac{1}{4}(1+\mathrm{i}\sqrt{3})\)
\( z=\frac{1}{4}(1-\mathrm{i}\sqrt{3})\)
\( z=-\frac{1}{2}\mathrm{i}\)
\( z=\frac{1}{2}(\cos 60^{\circ} - \mathrm{i} \sin 60^{\circ})\)
9000035709
Level:
A
Simplify \((1 -\mathrm{i})^{-3}\).
\(-\frac{1}
{4} + \frac{1}
{4}\mathrm{i}\)
\(1 + 3\mathrm{i}\)
\(- 2 - 2\mathrm{i}\)
\(\frac{1}
{2} + \frac{1}
{2}\mathrm{i}\)