# How do I find the negative power of a complex number?

Given a complex number of form $a + b i$,it can be proved that any power of it will be of the form $c + \mathrm{di}$.
For example, ${\left(a + b i\right)}^{2} = \left({a}^{2} - {b}^{2}\right) + 2 a b i$
${\left(a + b i\right)}^{-} n = \frac{1}{a + b i} ^ n = \frac{1}{c + \mathrm{di}}$
This final form is not acceptable, as it has a division by $i$, but we can use a factoring method to make it better.
$\left(m + n\right) \left(m - n\right) = {m}^{2} - {n}^{2}$ $\to$ $\left(m + n i\right) \cdot \left(m - n i\right) = {m}^{2} + {n}^{2}$
$\frac{1}{c + \mathrm{di}} \cdot \frac{\left(c - \mathrm{di}\right)}{\left(c - \mathrm{di}\right)} = \frac{c - \mathrm{di}}{{c}^{2} + {d}^{2}}$//