# How do you simplify (4 + 4i ) div(5 + 4i )?

Mar 9, 2016

$\frac{36}{41} + \frac{4}{41} i$

#### Explanation:

To simplify the fraction , require to make the denominator real.
This is achieved by multiplying the complex number on the denominator by it's $\textcolor{b l u e}{\text{ complex conjugate }}$

If (a + bi ) is a complex number then it's conjugate is (a - bi )

Note that the 'real part' remains unchanged , while the 'imaginary' part becomes negative.

and (a+bi)(a-bi) $= {a}^{2} - b i + b i - b {i}^{2} = {a}^{2} + {b}^{2} \text{ a real number }$

using $\left[{i}^{2} = {\left(\sqrt{- 1}\right)}^{2} = - 1\right]$

now multiply numerator and denominator by (5 - 4i )

$\Rightarrow \frac{\left(4 + 4 i\right) \left(5 - 4 i\right)}{\left(5 + 4 i\right) \left(5 - 4 i\right)} = \frac{20 - 16 i + 20 i - 16 {i}^{2}}{25 - 16 {i}^{2}}$

which simplifies to

$\frac{36 + 4 i}{41} = \frac{36}{41} + \frac{4}{41} i$