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So we have been given an assignment with four problems in each of four sections. I missed a day in class and was hoping you guys could help me out. There are four problems in each section.

  1. Use De Moivre's Theorem to find the powers of complex numbers below, and express the answer in the form $x + iy$

    $$(3 + 3i)^5$$

  2. Find the two square roots of

    $$1 + i \sqrt3$$

  3. Find the three cube roots of 64

  4. Find all the solutions to the equations

$$ x^2 + x + 1 = 0$$

Thank you.

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Imagine the complex numbers identified with the plane $\mathbb{R}^2$, by letting a complex number $a+bi$ correspond to the point $(a,b)$.

If we now translate to polar coordinates, so that $(a,b)$ corresponds to $(r,\theta)$ (where $a=r\cos\theta$ and $b=r\sin\theta$), then we can also express the number $a+bi$ as $$r(\cos\theta + i\sin\theta).$$ This, in turn, can be written as $$re^{i\theta},$$ since $e^{i\theta}=\cos\theta + i\sin\theta$ when $\theta$ is a real number.

De Moivre's Theorem says that if we write $a+bi = r(\cos\theta + i\sin\theta)$ as above, then $$(a+bi)^n = r^n\Bigl(\cos(n\theta) + i\sin(n\theta)\Bigr).$$ (An easy of seeing this is to use the complex exponential, since $$(a+bi)^n = (re^{i\theta})^n = r^n(e^{i\theta})^n = r^n e^{in\theta},$$ but this would require you to prove that the complex exponential has the same property as the real exponential that $(e^u)^v = e^{uv}$, so it might be a bit circular.)

More generally, if $a+bi = r\bigl(\cos\theta + i\sin\theta\bigr)$ and $c+di = s\bigl(\cos\phi + i\sin\phi)$, then $$(a+bi)(c+di) = rs\bigl(\cos(\theta+\phi) + i\sin(\theta+\phi)\bigr).$$ That is: you can multiply complex numbers by multiplying their norm (size) and adding their argument.

To solve the first problem, convert $3+3i$ into polar form, and use De Moivre's Theorem; then convert back.

To solve the second problem, convert $1+i\sqrt{3}$ into polar form, and figure out what numbers you need for $r$ and $\theta$ so that $r^2(\cos(2\theta) + i\sin(2\theta))$ equals that result. Similarly for $3$.

For the fourth problem, use the quadratic formula (and use a method like that of problem 2 to figure out the complex values of the square roots you get).

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