Question

Find the complex conjucate of (3+5i)/(1+2i)

Answer 1

$\frac{3+5i}{1+2i}$

We will multiply the numerator & denominator by conjugate of its denominator, to rationalise the denominator.

$(\frac{3+5i}{1+2i} ) (\frac{1-2i}{1-2i})$

$=\frac{3.1+3.(-2i)+5i.1+5i.(-2i)}{{(1)}^2-{(2i)}^2}$

$=\frac{3+(-6i)+5i+(-10{i}^2)}{1-4{i}^2}$
 
$=\frac{3-i-10{i}^2}{1-4{i}^2}$

[ i² = (√-1)² = -1 ]

$=\frac{3-i-10.(-1)}{1-4{i}^2}$

$=\frac{13-i}{5}$

Conjugate = $\frac{13+i}{5}$