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# De Moivre's Theorem

De Moivre's Theorem shows that to raise a complex number to the nth power, the absolute value is raised to the nth power and the argument is multiplied by n.

# Complex Numbers - Euler's Formula

Using Euler's formula $$e^{ix} = \cos x + i\sin x$$, evaluate

$\large e^{i \pi}.$

Which of the following is equivalent to the conjugate of the complex number $$4e^{i\pi /4}?$$

The complex number $$z = -4 + 3i$$ can be converted into the polar form $$z = re^{i\theta}.$$

What is the value of $$r?$$

If $$z = \sqrt{3} + i$$, then what is the value of $$z^6?$$

Hint: You might want to start by converting $$\sqrt{3} + i$$ into the form $$z = re^{i\theta}.$$

Given a complex number $$z = e^{ix}$$, for which of the following values of $$x$$ is the quantity $$\text{Re}(z)\text{Im(z)}$$ minimized?

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