Question

Represent the following identity in A+iB form : (2+3i)(5+4i) / (5+2i)(2+4i).​

Answer 1

GIVEN :–

$\\ \implies\bf \dfrac{(2+3i)(5+4i)}{(5+2i)(2+4i)}$

TO FIND :–

• Represent in A+iB form.

SOLUTION :–

• Let –

$\\ \implies\bf P = \dfrac{(2+3i)(5+4i)}{(5+2i)(2+4i)}$

$\\ \implies\bf P = \dfrac{(2 \times 5) + (2 \times 4i) + (3i \times 5) + (3i)(4i)}{(5 \times 2) + (5 \times 4i) + (2i \times 2) + (2i \times 4i)}$

$\\ \implies\bf P = \dfrac{10+8i+15i+12 {i}^{2} }{10+20i+4i+8 {i}^{2} }$

$\\ \implies\bf P = \dfrac{10+8i+15i - 12}{10+20i+4i - 8}$

$\\ \implies\bf P = \dfrac{ - 2 + 23i}{2+24i}$

• Now rationalization –

$\\ \implies\bf P = \dfrac{ - 2 + 23i}{2+24i} \times \dfrac{ 2 - 24i}{2 - 24i}$

$\\ \implies\bf P = \dfrac{ (- 2 + 23i)(2 - 24i)}{(2+24i)(2 - 24i)}$

$\\ \implies\bf P = \dfrac{ ( - 2 \times 2) + ( - 2 \times - 24i) + (23i \times 2) + (23i \times - 24i)}{ {(2)}^{2} - {(24i)}^{2}}$

$\\ \implies\bf P = \dfrac{ - 4+48i+46i+552}{54 + 576}$

$\\ \implies\bf P = \dfrac{94i+548}{630}$

$\\ \implies \large{ \boxed{\bf P = \dfrac{548}{630} + \dfrac{94}{630} i}}$