# How do you write (a+bi)/(c-di) in standard form?

Aug 20, 2016

$\frac{a + b i}{c - \mathrm{di}} = \left(\frac{a c - b d}{{c}^{2} + {d}^{2}}\right) + \left(\frac{a d + b c}{{c}^{2} + {d}^{2}}\right) i$

#### Explanation:

Multiply both numerator and denominator by the complex conjugate of the denominator, then simplify:

$\frac{a + b i}{c - \mathrm{di}}$

$= \frac{\left(a + b i\right) \left(c + \mathrm{di}\right)}{\left(c - \mathrm{di}\right) \left(c + \mathrm{di}\right)}$

$= \frac{a c + a \mathrm{di} + b c i + b {\mathrm{di}}^{2}}{{c}^{2} - {d}^{2} {i}^{2}}$

$= \frac{a c + a \mathrm{di} + b c i - b d}{{c}^{2} + {d}^{2}}$

$= \frac{\left(a c - b d\right) + \left(a d + b c\right) i}{{c}^{2} + {d}^{2}}$

$= \left(\frac{a c - b d}{{c}^{2} + {d}^{2}}\right) + \left(\frac{a d + b c}{{c}^{2} + {d}^{2}}\right) i$