# How do you simplify (2-4i)/(4-3i)?

Feb 10, 2016

$\frac{4}{5} - \frac{2}{5} i$

#### Explanation:

To simplify , require the denominator to be real. To achieve this multiply the numerator and denominator by the 'complex conjugate' of the denominator.

If (a + bi ) is a complex number then (a - bi ) is the conjugate.

Note that : (a + bi )(a - bi ) = ${a}^{2} + {b}^{2} \textcolor{b l a c k}{\text{ which is real }}$

here the conjugate of (4 - 3i ) is (4 + 3i )

hence : $\frac{\left(2 - 4 i\right) \left(4 + 3 i\right)}{\left(4 - 3 i\right) \left(4 + 3 i\right)}$

distribute using FOIL

$= \frac{8 + 6 i - 16 i - 12 {i}^{2}}{16 + 12 i - 12 i - 9 {i}^{2}}$

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

$= \frac{8 - 10 i + 12}{16 + 9} = \frac{20 - 10 i}{25} = \frac{20}{25} - \frac{10}{25} i = \frac{4}{5} - \frac{2}{5} i$