Question

(\frac{6 + 6i}{4 + 3i})

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $4-3i$:
$\frac{(6+6i)(4-3i)}{(4+3i)(4-3i)}$

Step2: Expand numerator

Use FOIL method for numerator:
$(6)(4) + (6)(-3i) + (6i)(4) + (6i)(-3i) = 24 - 18i + 24i - 18i^2$
Simplify using $i^2=-1$:
$24 + 6i - 18(-1) = 24 + 6i + 18 = 42 + 6i$

Step3: Expand denominator

Use difference of squares:
$4^2 - (3i)^2 = 16 - 9i^2$
Simplify using $i^2=-1$:
$16 - 9(-1) = 16 + 9 = 25$

Step4: Split into real/imaginary parts

Write the fraction as two terms:
$\frac{42}{25} + \frac{6}{25}i$

Answer:

$\frac{42}{25} + \frac{6}{25}i$