How do you divide (-3+10i)/(-6i)?

Oct 15, 2016

$\frac{- 10 - 3 i}{6} = - \frac{5}{3} - \frac{1}{2} i$

Explanation:

$\frac{- 3 + 10 i}{- 6 i}$

Multiply by the conjugate of the denominator over itself.
A conjugate of a complex number $a + b i$ is $a - b i$.
The denominator can be written as $0 - 6 i$, so the conjugate is $0 + 6 i$ or $6 i$

$\frac{\left(- 3 + 10 i\right)}{- 6 i} \cdot \frac{6 i}{6 i}$

Distribute in the numerator and multiply in the denominator.

$\frac{- 3 \cdot 6 i + 10 i \cdot 6 i}{- 36 {i}^{2}}$

$\frac{- 18 i + 60 {i}^{2}}{- 36 {i}^{2}}$

Recall that ${i}^{2} = - 1$

$\frac{- 18 i + 60 \left(- 1\right)}{- 36 \left(- 1\right)}$

$\frac{- 18 i - 60}{36}$

Factor out a 6 in both numerator and denominator.

$\frac{6 \left(- 3 i - 10\right)}{6 \left(6\right)}$

$\frac{\cancel{6} \left(- 3 i - 10\right)}{\cancel{6} \left(6\right)}$

$\frac{- 3 i - 10}{6}$ or

$\frac{- 10 - 3 i}{6} = - \frac{10}{6} - \frac{3 i}{6} = - \frac{5}{3} - \frac{i}{2} = - \frac{5}{3} - \frac{1}{2} i$