Aus Online Mathematik Brückenkurs 2

One way to determine the magnitude is to calculate the product \displaystyle \left( \text{3}-\text{4}i \right)\left( \text{3}+\text{2}i \right) and then to take the magnitude of the result, but for products

\displaystyle \left| zw \right|=\left| z \right|\centerdot \left| w \right|

and we can take the magnitude of the factors \displaystyle \text{3}-\text{4}i and \displaystyle \text{3}+\text{2}i and then multiply the magnitudes together:

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