Question

Simplify the given and express in the form of a + ib.
$\frac{5+7i}{4+3i} +\frac{5+7i}{4-3i}$

Answer 1

Solution:

To express the given expression in the form of a + ib ,take LCM

$\frac{5+7i}{4+3i} +\frac{5+7i}{4-3i} \\ \\ = \frac{(5 + 7i)(4 - 3i) + (5 + 7i)(4 + 3i)}{(4 + 3i)(4 - 3i) } \\ \\ \frac{20 - 15i + 28i - 21 {i}^{2} + 20 + 15i + 28i + 21 {i}^{2} }{16 +9 } \\ \\ = \frac{40 + 56i + 21 - 21}{25} \\ \\ = \frac{40 + 56i}{25} \\ \\ a + ib = \frac{40 }{25} + i \frac{56}{25} \\ \\ a + ib = \frac{8 }{5} + i \frac{56}{25} \\ \\so \\ \\ a = \frac{8}{5} \\ \\ b = \frac{56}{25}$
Hope it helps you.