# How do you divide  (6-i) / (7-2i) ?

Apr 8, 2016

$\frac{44}{53} + \frac{5}{53} i$

#### Explanation:

To divide this fraction we require to rationalise the denominator.

We do this by multiplying numerator/denominator by the$\textcolor{b l u e}{\text{ complex conjugate " " of the denominator }}$

If $\textcolor{b l u e}{\text{ a ± bi " " is a complex number then }}$

$\textcolor{red}{\text{ a ∓ bi " " is it's conjugate }}$

Note that the 'real part' remains unchanged , while the sign of the 'imaginary part' changes.

Also (a+ bi)(a - bi) = ${a}^{2} - {b}^{2} \text{ a real number }$

and ${i}^{2} = {\left(\sqrt{- 1}\right)}^{2} = - 1$

Now the conjugate of 7 - 2i is 7 + 2i

multiplying numerator / denominator by (7 + 2i)

$\Rightarrow \frac{\left(6 - i\right) \left(7 + 2 i\right)}{\left(7 - 2 i\right) \left(7 + 2 i\right)} = \frac{42 + 5 i + 2}{49 + 4} = \frac{44}{53} + \frac{5}{53} i$