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I am presented with the following problem:

Solve the following equation: $$ z^4 = −4 $$

The z in this question represents a complex number with 4 complex roots, which I am supposed to find.

I started by putting it to polar form, with magnitude equal to $|z^4| = \sqrt{(-4)^2+0^2}= 4$ and angle of 180 degrees, or π [rad]. Then I used general formula for exponential form $ z = |z|e^{iθ}$ and empowered it by power of 4 so I get $ z^4 = |z|^4e^{4iθ}$.

There I got stuck and I have no clue whatsoever how to solve this. Any help would be appreciated.

The answers I should get to are: $1 + i$; $1 - i$; $ -1 + i$ ; $ -1 - i $.

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    I have not studied De Moivre's formula yet, and therefore I didn't use it. The solution should be obtainable with Euler's formula though. – 2017-01-15
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    What about $e^{i\theta}=\cos \theta +i \sin \theta$? – 2017-01-15
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    Could you please elaborate? I've been trying to use it multiple times reaching e^{i*π/4} = 4 * (cos π/4 + i*sin π/4), which is indeed not the answer I should have got. – 2017-01-15

2 Answers 2

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$$ z^4+4 = (z^2+2i)(z^2-2i) $$

Notice that ${(e^{i\pi/4})}^2 = e^{i\pi/2}=i$ so $$ z^2-2i = (z+\sqrt2e^{i\pi/4})(z-\sqrt2e^{i\pi/4}) $$

And similarly $$ z^2+2i = (z+\sqrt2ie^{i\pi/4})(z-\sqrt2ie^{i\pi/4}) $$

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We have $z^4=-4$. Write $z=|z|e^{i\theta}$ and $w = -4 = 4e^{i \pi}$, so $|z|^4=|w|$ and thus $|z|=\sqrt{2}$. Furthermore, $e^{4i\theta}=e^{i\pi}$, so $4\theta=\pi +\frac{2\pi k}{4}$, $k=0,1,2,3$.

Then use $e^{i\theta}=\cos \theta + i \sin \theta$ for $k=0,1,2,3$.

For example, if $k=0$, then $\theta = \pi /4$, hence $z = |z|e^{i\theta} = \sqrt{2}e^{\pi / 4}=\sqrt{2}(\cos{\pi / 4}+i \sin{\pi / 4})=\sqrt{2}(\frac{\sqrt{2}}{2}+i \frac{\sqrt{2}}{2})=1+i$.

If $k=1$, then $\theta=\pi/4+\pi/2$, hence $z=-1+i$...

For $k=2$ you should get $-1-i$ and for $k=3$ you get $1-i$.