Complex number argument and module

[amiras]

Homework Statement

$$z=\frac{(1-3i)^{100} * i * (7-5i)}{5+7i}$$

Homework Equations

Find module and argument of z.

The Attempt at a Solution

Assuming:

$$|z_1 * z_2| = |z_1||z_2|$$

$$|z|=\frac{|1-3i|^{100} * |i| * |7-5i|}{|5+7i|}$$

And now calculate each module individually by "Pythagoras theorems"

$$|z|=\frac{\sqrt{10}^{100} * 1 *\sqrt{74}}{\sqrt{74}}=10^{50}$$And I do now know how to calculate argument now.

 

[danago]

You could express each complex number in polar form and then simplify it using the properties of complex numbers expressed in polar form:

$$\begin{array}{l} {r_1}cis({\theta _1}) \times {r_2}cis({\theta _2}) = {r_1}{r_2}cis({\theta _1} + {\theta _2})\\ {\left( {rcis(\theta )} \right)^n} = {r^n}cis(n\theta ) \end{array}$$