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In Fraleigh's A First Course in Abstract Algebra he solves the problem

$z^4=-16$

by rewriting in polar form that gives

$z^4=|z|^4(\cos(4\theta)+i\sin(4\theta))$
$|z|^4(\cos(4\theta)+i\sin(4\theta))=-16+0i = 16(-1+0i)$

He then says that we can immediately conclude that $|z|^4=16$. He does not go through the process of taking the absolute value of $z$. Assuming that it is $16$ just because it is in the right form seems strange to me. If you can conclude this based on form then why couldn't you factor out some other number like $-16$?

Jac Frall
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1 Answers1

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Well $|z^4|=|z|^4=|-16|=16$ is pretty obvious because when you take the modulus you are dealing with non-negative real numbers.

Then $\cos^2 4\theta +\sin ^2 4\theta =1$ which is a feature of the polar form and is replicated in the form $|e^{i\phi}|=1$ for any real $\phi$ - we take $\phi = 4\theta$ here for convenience because we know we will be taking a fourth root.

Essentially, then, this is putting $z=|z|e^{i\theta}$ so that $z^4=|z|^4 e^{i4\theta}$, and this is a standard technique for such questions. Some care needs to be taken, however you do this, to identify all four of the fourth roots.

Mark Bennet
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