# How do you simplify (4-4i)/(5+3i)?

Nov 21, 2015

$\frac{2}{9} - \frac{8}{9} i$

#### Explanation:

Use the conjugate of the denominator. (Recall that the conjugate of $a + b$ is $a - b$.)

We can multiply the expression by $\frac{5 - 3 i}{5 - 3 i}$.

$\frac{4 - 4 i}{5 + 3 i} \left(\frac{5 - 3 i}{5 - 3 i}\right) = \frac{20 - 20 i - 12 i + 12 {i}^{2}}{25 \cancel{- 15 i + 15 i} - 9 {i}^{2}} = \frac{20 - 32 i + 12 {i}^{2}}{25 - 9 {i}^{2}}$

Recall that $i = \sqrt{1}$, so ${i}^{2} = - 1$.

$\frac{20 - 32 i + 12 \left(- 1\right)}{25 - 9 \left(- 1\right)} = \frac{20 - 12 - 32 i}{25 + 9} = \frac{8 - 32 i}{36} = \frac{2}{9} - \frac{8}{9} i$

Note that the answer is written in the complex number format $a + b i$.