Aus Online Mathematik Brückenkurs 2

For magnitudes of quotients, we have the arithmetical rule

\displaystyle \left| \frac{z}{w} \right|=\frac{\left| z \right|}{\left| w \right|}

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