Question

Find each principle root. Express the result in the form a+bi with a and b rounded to the nearest tenth: (2+2i)^1/3

Answer 1

Where are you stuck?

Answer 2

I only went up to \[(2\sqrt{2})^{1/3}(\cos45+360(0)+isin46+360(0))\]

Answer 3

Let z= 2+2i, then we need find third roots of it.
\(|z|=2\sqrt2\)

Answer 4

and \(z=|z|e^{i\theta}\)

Answer 5

Let \(\omega\) be roots of z, and \(\omega =|\omega| e^{i\tau}\)
\(\omega^3=|\omega|^3e^{3i\tau}\)

Answer 6

\(|\omega|^3= 2\sqrt 2\), hence \(|\omega| =\sqrt[6]8\)

Answer 7

\(e^{3i\tau}= e^{i\theta}\), then \(3\tau = \theta +2\pi n\), n =0,1,2
That gives us \(\tau = \theta /3+2\pi n/3\)

Answer 8

what is \(\theta\)?

Answer 9

Hence \(\theta =\pi/4\)