# How do you simplify (10+i)/(4-i)?

Oct 9, 2016

In order to remove the complex denominator, multiply the top and bottom by the complex conjugate of the dominator you want to eliminate, thus:

$\frac{10 + i}{4 - i} = \frac{10 + i}{4 - i} . \frac{4 + i}{4 + i}$
$= \frac{\left(10 + i\right) \left(4 + i\right)}{\left(4 - i\right) \left(4 + i\right)}$

Then multiply out the denominator and the numerator; the denominator will now be real:

$\therefore \frac{10 + i}{4 - i} = \frac{\left(10\right) \left(4\right) + \left(10\right) \left(i\right) + \left(4\right) \left(i\right) + \left(i\right) \left(i\right)}{\left(4\right) \left(4\right) + \left(4\right) \left(- i\right) + \left(4\right) \left(i\right) + \left(i\right) \left(- i\right)}$
$= \frac{40 + 10 i + 4 i + {i}^{2}}{16 + 4 i - 4 i - {i}^{2}}$
$= \frac{40 + 14 i - 1}{16 + 1}$
$= \frac{39 + 14 i}{17}$