# How do you change 10z^4 into polar coordinates where z = 2 +4i?

May 23, 2017

$4000 \left(\cos \theta + i \sin \theta\right)$

#### Explanation:

Let's square it two times.

${z}^{2} = 4 + 16 i - 16$

${z}^{4} = {\left(- 12 + 16 i\right)}^{2} = 144 - 24 \cdot 16 i - 256 = - 112 - 384 i$

$10 {z}^{4} = - 1120 - 3840 i = w$

$| w | = \sqrt{{1120}^{2} + {3840}^{2}} = 4000$

$\tan \left(\textrm{a r g} w\right) = \frac{3840}{1120} = \frac{24}{7}$

$\theta = \textrm{a r g} w = \pi + \arctan \frac{24}{7}$