Question

Simplify the following and express in the form a + ib:

${\displaystyle \frac{5+7\text{i}}{4+3\text{i}}+\frac{5+7\text{i}}{4-3\text{i}}}$

Answer 1

${\displaystyle \frac{5+7\text{i}}{4+3\text{i}}+\frac{5+7\text{i}}{4-3\text{i}}}$

= ${\displaystyle (5+7\text{i})[\frac{1}{4+3\text{i}}+\frac{1}{4-3\text{i}}]}$

= ${\displaystyle (5+7\text{i})\left[\frac{4-3\text{i}+4+3\text{i}}{(4+3\text{i})(4-3\text{i})}\right]}$

= ${\displaystyle (5+7\text{i})\left[\frac{8}{16-9{\text{i}}^2}\right]}$

= ${\displaystyle (5+7\text{i})\left[\frac{8}{16-9(-1)}\right]}$      ...[∵ i² = – 1]

= ${\displaystyle \frac{8(5+7\text{i})}{25}}$

= ${\displaystyle \frac{40+56\text{i}}{25}}$

= ${\displaystyle \frac{40}{25}+\frac{56}{25}\text{i}}$

= ${\displaystyle \frac{8}{5}+\frac{56}{25}\text{i}}$.