Solution to the following complex equation

Question

I need to find the solution to the following complex equation: $z^4 = −8 + 8\sqrt3i$

I know that I may write:

$z^4 = [ρ^4(cos 4θ + isin 4θ)]= −8 + 8\sqrt3i$

But then I do not know how to proceed. Possibly I would like to find a method which is the most systematic. Can you help me?

Answer 1

Note,

$$−8 + 8\sqrt3i=16\left( -\frac12 + i \frac{\sqrt3}2\right) = 16\left[ \cos\left(\frac{2\pi}3+2\pi n\right)+ i \sin\left(\frac{2\pi}3+2\pi n\right)\right] $$

Compare with $z^4 = ρ^4(\cos 4θ + i\sin 4θ)$ to obtain

$$ z = 2 \cos\left(\frac{\pi}6+\frac{\pi n}2\right)+ i 2\sin\left(\frac{\pi}6+\frac{\pi n}2\right)$$

Answer 2

If $z = \rho e^{i\theta}$ then $z^4 = \rho^4 e^{4 i \theta}$. Find what values of $\rho$ and $\theta$ will make this equal to $-8 + 8\sqrt{3} i$. (For instance, $\rho^4$ should be the modulus of $-8+8\sqrt{3}i$, etc.)

Answer 3

I agree with the peterwhy's comment and angryavian's answer. However, I would like to address why the OP is having trouble with this problem. In my opinion, either the problem is unfair or it is not unfair.

If the OP has not taken any course in complex analysis, and not consulted any complex analysis textbook, then I regard the problem as unfair. The OP can't be expected to attack this type of problem without first developing the intuition that comes from attacking simple complex analysis problems. In fact, if complex analysis is new to the OP, then I would advise the OP to avoid attacking complex analysis problems in a disorganized fashion and instead (for example) find a complex analysis textbook, and begin on page 1.

However, the OP's query suggests that he has a nodding acquaintance to basic complex analysis principles, without any corresponding intuition developed. I wonder how the OP came by this nodding acquaintance. If it is self-taught, then I think the opinion in my previous paragraph pertains. If instead, the OP is in the middle of a complex analysis course, then I regard the question as fair, but caution the OP that he is headed for trouble (i.e. without developing intuition, the OP will typically be lost re similar problems). I advise the OP to go back to his teacher, explain his lack of developed intuition, and ask the teacher to help him develop his intuition. As indicated in the previous paragraph, this intuition is normally developed by attacking the pertinent textbook, starting on page 1.

Normally I would have downvoted, because the OP didn't show any work. This time I didn't downvote, because you can't meaningfully show work without intuition, which the OP seems to be lacking.