Answer
$\displaystyle \frac{ac-bd}{c^{2}+d^{2}}+\frac{ad+bc}{c^{2}+d^{2}}i$
Work Step by Step
$\displaystyle \frac{a+bi}{c-di}\qquad$ ...rationalize by multiplying both the numerator and denominator with $c+di$.
$=\displaystyle \frac{(a+bi)(c+di)}{(c-di)(c+di)}$
$\qquad$ ...use the FOIL method in the numerator
and the difference of squares: $(a-b)(a+b)=a^{2}-b^{2}$ in the denominator.
$=\displaystyle \frac{ac+adi+bci+bdi^{2}}{c^{2}-(di)^{2}}\qquad$ ...simplify and add like terms ($i^{2}=-1$).
$=\displaystyle \frac{ac+adi+bci-bd}{c^{2}+d^{2}}\qquad$ ...group the real and imaginary parts in the numerator.
$=\displaystyle \frac{ac-bd+(ad+bc)i}{c^{2}+d^{2}}\qquad$ ...write in standard form $a+bi$
$=\displaystyle \frac{ac-bd}{c^{2}+d^{2}}+\frac{ad+bc}{c^{2}+d^{2}}i$
