Aus Online Mathematik Brückenkurs 2

For magnitudes of quotients, we have the arithmetical rule

\displaystyle \left|\frac{z}{w}\right| = \frac{|z|}{|w|}\,\textrm{.}

We can therefore take the magnitude of the numerator and denominator separately and then divide the magnitudes by each other,

\displaystyle \begin{align} \left|\frac{3-4i}{3+2i}\right| &= \frac{|3-4i|}{|3+2i|} = \frac{\sqrt{3^2+(-4)^2}}{\sqrt{3^2+2^2}} = \frac{\sqrt{9+16}}{\sqrt{9+4}} = \frac{\sqrt{25}}{\sqrt{13}} = \frac{5}{\sqrt{13}}\,\textrm{.} \end{align}