Question

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

Answer 1

`(-4sqrt(2)+4sqrt(2)i)/(6+6i) = 2/3sqrt(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+6i = 6(1+i)`

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

`(-4sqrt(2)+4sqrt(2)i)/(6+6i) = (-4sqrt(2)(1-i)(1-i))/(6(1+i)(1-i))`

`color(white)((-4sqrt(2)+4sqrt(2)i)/(6+6i)) = (-4sqrt(2)(1-2i+i^2))/(6(1-i^2))`

`color(white)((-4sqrt(2)+4sqrt(2)i)/(6+6i)) = (8sqrt(2)i)/12`

`color(white)((-4sqrt(2)+4sqrt(2)i)/(6+6i)) = 2/3sqrt(2)i`