Question

division of complex numbers
ex: $(2 - 4i) \div (8 - 6i)$

Explanation:

Step1: Write as fraction, simplify

First, rewrite division as a fraction, then factor numerator and denominator to simplify:
$\frac{2-4i}{8-6i} = \frac{2(1-2i)}{2(4-3i)} = \frac{1-2i}{4-3i}$

Step2: Multiply by conjugate

Multiply numerator and denominator by the conjugate of the denominator ($4+3i$):
$\frac{(1-2i)(4+3i)}{(4-3i)(4+3i)}$

Step3: Expand numerator

Use FOIL to expand the numerator:
$(1)(4) + (1)(3i) + (-2i)(4) + (-2i)(3i) = 4 + 3i - 8i -6i^2$
Since $i^2=-1$, substitute:
$4 -5i -6(-1) = 4 -5i +6 = 10 -5i$

Step4: Expand denominator

Use difference of squares for the denominator:
$4^2 - (3i)^2 = 16 -9i^2$
Substitute $i^2=-1$:
$16 -9(-1) = 16 +9 =25$

Step5: Simplify the fraction

Divide the numerator by the denominator:
$\frac{10-5i}{25} = \frac{10}{25} - \frac{5i}{25} = \frac{2}{5} - \frac{1}{5}i$

Answer:

$\frac{2}{5} - \frac{1}{5}i$