Question

simplify the following expression.
\\(\frac{4i}{4 + 2i}\\)
answer

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $4-2i$:
$\frac{4i}{4+2i} \times \frac{4-2i}{4-2i} = \frac{4i(4-2i)}{(4+2i)(4-2i)}$

Step2: Expand numerator

Use distributive property:
$4i(4-2i) = 16i - 8i^2$
Since $i^2=-1$, substitute:
$16i - 8(-1) = 8 + 16i$

Step3: Expand denominator

Use difference of squares:
$(4+2i)(4-2i) = 4^2 - (2i)^2 = 16 - 4i^2$
Substitute $i^2=-1$:
$16 - 4(-1) = 20$

Step4: Simplify the fraction

Divide numerator terms by denominator:
$\frac{8 + 16i}{20} = \frac{8}{20} + \frac{16i}{20}$
Reduce fractions:
$\frac{2}{5} + \frac{4}{5}i$

Answer:

$\frac{2}{5} + \frac{4}{5}i$