+116 Cant figure this one out:
Suppose \(z\) is a complex number such that \(z^3 = 100+75i\) . Find \(|z|\).
+118673 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,
\(|z|=5\)
