# How do you simplify (-4sqrt2+4sqrt2i)div(6+6i)?

Nov 8, 2017

$\frac{- 4 \sqrt{2} + 4 \sqrt{2} i}{6 + 6 i} = \frac{2}{3} \sqrt{2} i$

#### Explanation:

Normally when rationalising the quotient of two complex numbers you would multiply numerator and denominator by the compelx conjugate of the denominator.

In this example, the denominator is:

$6 + 6 i = 6 \left(1 + i\right)$

So instead of multiplying by the complex conjugate $6 - 6 i$ and introducing an unnecessary extra factor $6$, we can just multiply both by $\left(1 - i\right)$ as follows:

$\frac{- 4 \sqrt{2} + 4 \sqrt{2} i}{6 + 6 i} = \frac{- 4 \sqrt{2} \left(1 - i\right) \left(1 - i\right)}{6 \left(1 + i\right) \left(1 - i\right)}$

$\textcolor{w h i t e}{\frac{- 4 \sqrt{2} + 4 \sqrt{2} i}{6 + 6 i}} = \frac{- 4 \sqrt{2} \left(1 - 2 i + {i}^{2}\right)}{6 \left(1 - {i}^{2}\right)}$

$\textcolor{w h i t e}{\frac{- 4 \sqrt{2} + 4 \sqrt{2} i}{6 + 6 i}} = \frac{8 \sqrt{2} i}{12}$

$\textcolor{w h i t e}{\frac{- 4 \sqrt{2} + 4 \sqrt{2} i}{6 + 6 i}} = \frac{2}{3} \sqrt{2} i$