Question

find modules of 3/2-4i

Answer 1

$z = \frac{3}{2 - 4i} \\ = \frac{3}{2 - 4i} \times \frac{2 + 4i}{2 + 4i} \\ = \frac{3(2 + 4i)}{ {2}^{2} - {(4i)}^{2} } \\ = \frac{6 + 12i}{4 - 16 {i}^{2} } \\ =\frac{6 + 12i}{4 - 16 ( - 1)} \\ = \frac{6 + 12i}{4 + 16} \\ = \frac{6 + 12i}{20} \\ \\ |z| = \sqrt{( { \frac{6}{20} })^{2} + { (\frac{12}{20}) }^{2} } \\ = \frac{1}{20} \sqrt{ {6}^{2} + {12}^{2} } \\ = \frac{1}{20} \sqrt{36 + 144} \\ = \frac{1}{20} \sqrt{180} \\ = \frac{1}{20} \sqrt{2 \times 2 \times 3 \times 3 \times 5} \\ = \frac{2 \times 3}{20} \sqrt{5} \\ = \frac{3}{10} \sqrt{5}$