Dividing Complex Numbers

$\frac{8}{1+i}$

$\frac{8}{1+i}•\frac{(1-i)}{(1-i)}$

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)

$\frac{8-8i}{1-i+i-i^2}$

use distributive property to eliminate parentheses

$\frac{8-8i}{1-i^2}$

combine like terms in the denominator

$\frac{8-8i}{1-(-1)}$

replace imaginary numbers with exponents with the simplest form

(remember that i² = -1)

$\frac{8-8i}{2}$

simplify

$\frac{8}{2}-\frac{8i}{2}$

write in a + bi form

4 - 4i

reduce fractions if possible

$\frac{3+i}{3-i}$

$\frac{3+i}{3-i}•\frac{(3+i)}{(3+i)}$

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)

$\frac{9+3i+3i+i^2}{9+3i-3i-i^2}$

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

( i² = -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