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

Aug 12, 2016

$\frac{3}{5} + \frac{4}{5} i$

#### Explanation:

To simplify this fraction, we require the denominator to be real.

To achieve this multiply both the numerator and denominator by the $\textcolor{b l u e}{\text{complex conjugate}}$ of the denominator.

Given z = a ± bi then the conjugate is color(red)(bar(z))=color(red)(a)∓color(red)(b)i

Note that the real part remains unchanged while the sign of the imaginary part is reversed.

Also $\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\left(a + b i\right) \left(a - b i\right) = {a}^{2}} \textcolor{w h i t e}{\frac{a}{a}} |}}} \text{ the result is real}$
where a and b are real.

$\Rightarrow 3 - 4 i \text{ has conjugate } 3 + 4 i$

multiply numerator/denominator by (3+4i)

$\Rightarrow \frac{5 \left(3 + 4 i\right)}{\left(3 - 4 i\right) \left(3 + 4 i\right)} = \frac{15 + 20 i}{25} = \frac{3}{5} + \frac{4}{5} i$