Cant figure this one out:

Suppose \(z\) is a complex number such that \(z^3 = 100+75i\) . Find \(|z|\).

somebody Oct 13, 2018

#7**+2 **

Suppose z is a complex number such that \(z^3 = 100+75i\) . Find |z|

\(|z^3| = \sqrt{100^2+75^2}\\ |z^3| = \sqrt{100^2+75^2}\\ |z^3| = 125\\ z^3=125[\frac{100+75i}{125}]\\ z^3=125[\ 0.8+0.5i]\\ \)

Let

\(z=re^{i\theta}\\ z^3=r^3e^{3i\theta}\\ z^3=r^3(cos(3\theta) +isin(3\theta))\\ \)

\(r^3(cos(3\theta) +isin(3\theta))=125[\ 0.8+0.6i]\\ r^3=125\\ r=5\\ cos(3\theta)=0.8\\ sin(3\theta)=0.6\\ 3\theta \approx 0.6435\\ \theta \approx 0.2145\;radians\\ \)

anyway, I digress,

\(|z|=5\)

Melody Oct 14, 2018