# How do you write the following quotient in standard form 4/(4-5i)?

Feb 7, 2017

$\frac{16}{41} + \frac{20}{41} i$

#### Explanation:

To perform the division, we require to multiply the numerator/denominator by the $\textcolor{b l u e}{\text{complex conjugate}}$ of the denominator. This ensures that we have a rational value on the denominator.

The conjugate of $4 - 5 i \text{ is } 4 \textcolor{red}{+} 5 i$

$\Rightarrow \frac{4}{4 - 5 i} = \frac{4 \left(4 + 5 i\right)}{\left(4 - 5 i\right) \left(4 + 5 i\right)} = \frac{16 + 20 i}{41}$

$\textcolor{b l u e}{\text{Evaluating denominator}}$

$\rightarrow \left[\left(4 - 5 i\right) \left(4 + 5 i\right) = 16 - 20 i + 20 i - 25 {i}^{2} = 41\right] \leftarrow$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} {i}^{2} = {\left(\sqrt{- 1}\right)}^{2} = - 1$

$\Rightarrow \frac{4}{4 - 5 i} = \frac{16 + 20 i}{41} = \frac{16}{41} + \frac{20}{41} i$