# How do you write the complex conjugate of the complex number (6-3i)/ (2+i)?

Nov 27, 2016

The complex conjugate is $= \frac{9}{5} + \frac{12}{5} i$

#### Explanation:

If you want to simplify a quotient of complex numbers , multiply numerator and denominator by the conjugate of the denominator.

$z = {z}_{1} / {z}_{2} = \frac{{z}_{1} {\overline{z}}_{2}}{{z}_{2} {\overline{z}}_{2}}$

Also, $\overline{z} = {\overline{z}}_{1} / {\overline{z}}_{2}$

The conjugate of $\left(a + i b\right)$ is $\left(a - i b\right)$

${i}^{2} = - 1$

So,

$\frac{6 - 3 i}{2 + i} = \frac{3 \left(2 - i\right)}{2 + i}$

$= \frac{3 \left(2 - i\right) \left(2 - i\right)}{\left(2 + i\right) \left(2 - i\right)}$

$= \frac{3 \left(4 - 4 i + {i}^{2}\right)}{4 - {i}^{2}}$

$= \frac{3 \left(3 - 4 i\right)}{5}$

$= \frac{9}{5} - \frac{12}{5} i$

So, the complex conjugate is $= \frac{9}{5} + \frac{12}{5} i$

Second way,

The conjugate is $\frac{\overline{6 - 3 i}}{\overline{2 + i}}$

$= \frac{6 + 3 i}{2 - i} = \frac{\left(6 + 3 i\right) \left(2 + i\right)}{\left(2 - i\right) \left(2 + i\right)}$

$= \frac{12 + 12 i + 3 {i}^{2}}{4 - {i}^{2}}$

$= \frac{9 + 12 i}{5}$

$= \frac{9}{5} + \frac{12 i}{5}$