Dividing Complex Numbers
Example 1:
$\frac{8}{1+i}$ 

$\frac{8}{1+i}\u2022\frac{(1i)}{(1i)}$ 
multiply numerator and denominator by the complex conjugate of the denominator The complex conjugate of 1 + i is 1  i (change the sign in the middle) 
$\frac{88i}{1i+i{i}^{2}}$ 
use distributive property to eliminate parentheses 
$\frac{88i}{1{i}^{2}}$ 
combine like terms in the denominator 
$\frac{88i}{1(1)}$ 
replace imaginary numbers with exponents with the simplest form (remember that i^{2} = 1) 
$\frac{88i}{2}$ 
simplify 
$\frac{8}{2}\frac{8i}{2}$ 
write in a + bi form 
4  4i 
reduce fractions if possible 
$\frac{3+i}{3i}$ 

$\frac{3+i}{3i}\u2022\frac{(3+i)}{(3+i)}$ 
multiply numerator and denominator by the complex conjugate of the denominator The complex conjugate of 3  i is 3 + i (change the sign in the middle) 
$\frac{9+3i+3i+{i}^{2}}{9+3i3i{i}^{2}}$ 
use distributive property to eliminate parentheses 
$\frac{9+6i+{i}^{2}}{9{i}^{2}}$ 
Simplify numerator and denominator separately 
$\frac{9+6i+(1)}{9(1)}$ 
replace imaginary numbers with exponents with the simplest form ( i^{2} = 1) 
$\frac{8+6i}{10}$ 
simplify 
$\frac{8}{10}+\frac{6i}{10}$ 
write in a + bi form 
$\frac{4}{5}+\frac{3}{5}i$ 
reduce fractions if possible 
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