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

May 23, 2016

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

#### Explanation:

When dividing complex numbers , to simplify we rationalise the denominator. This means changing it into a rational number instead of a complex number.

This is achieved by multiplying the numerator and denominator by the $\textcolor{b l u e}{\text{ conjugate of the denominator}}$

Given a complex number a + bi , conjugate is a - bi

Note that a and b remain unchanged whist the sign changes.

In general conjugate of a ± bi is a ∓ bi

here the conjugate of 4 - 2i is 4 + 2i

$\Rightarrow \frac{4 + 2 i}{4 - 2 i} \times \frac{4 + 2 i}{4 + 2 i} = \frac{\left(4 + 2 i\right) \left(4 + 2 i\right)}{\left(4 - 2 i\right) \left(4 + 2 i\right)}$

expanding numerator and denominator.

$= \frac{16 + 16 i + 4 {i}^{2}}{16 - 4 {i}^{2}}$

Using $\left({i}^{2} = - 1\right)$ this now becomes

$\frac{16 + 16 i - 4}{16 + 4} = \frac{12 + 16 i}{20} = \frac{12}{20} + \frac{16}{20} i = \frac{3}{5} + \frac{4}{5} i$