How do you simplify 3/(6-3i)?

Oct 20, 2016

$\frac{2}{5} + \frac{1}{5} i$

Explanation:

To simplify this fraction, we must multiply the numerator/denominator by the $\textcolor{b l u e}{\text{complex conjugate}}$ of the denominator.
This ensures that the denominator is a real number.

Given a complex number z=x±yi then the complex conjugate is.

color(red)(bar(ul(|color(white)(2/2)color(black)(barz=x∓yi)color(white)(2/2)|)))

Note that the real part remains unchanged while the $\textcolor{red}{\text{sign}}$ of the imaginary part is reversed.

Hence the conjugate of $6 - 3 i \text{ is } 6 + 3 i$

and $\left(6 - 3 i\right) \left(6 + 3 i\right) = 36 + 9 = 45 \leftarrow \text{ a real number}$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{{i}^{2} = {\left(\sqrt{- 1}\right)}^{2} = - 1} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

Multiply numerator/denominator by 6 + 3i

$\frac{3 \left(6 + 3 i\right)}{\left(6 - 3 i\right) \left(6 + 3 i\right)} = \frac{18 + 9 i}{45} = \frac{18}{45} + \frac{9}{45} i = \frac{2}{5} + \frac{1}{5} i$