How do you divide 1/(-8-5i)?

Sep 8, 2016

Answer:

$- \frac{8}{89} + \frac{5}{89} i$

Explanation:

To divide this fraction we require to make the denominator real.

This can be achieved by multiplying (-8 - 5i) by it's $\textcolor{b l u e}{\text{conjugate}}$

If z=x±yi then the conjugate is.

color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(barz=x∓yi)color(white)(a/a)|)))
Note that x, remains unchanged while the sign of the imaginary part is reversed.

$\Rightarrow \left(- 8 + 5 i\right) \text{ is the conjugate of} - 8 - 5 i$

and $\left(- 8 - 5 i\right) \left(- 8 + 5 i\right) = 64 - 25 {i}^{2} = 64 + 25 = 89$

That is we have a real value on the denominator.

Since this is a fraction, we must of course multiply both numerator and denominator by the conjugate.

$\Rightarrow \frac{- 8 + 5 i}{\left(- 8 - 5 i\right) \left(- 8 + 5 i\right)} = \frac{- 8 + 5 i}{89} = - \frac{8}{89} + \frac{5}{89} i$