Question

review the graphs of complex numbers z and w. what is \\(\frac{z}{w}\\)? \\(\circ -4 + 16i\\) \\(\circ 6 + 10i\\) \\(\circ 7 + 11i\\) \\(\circ 8 + 2i\\)

Explanation:

Step1: Identify complex numbers

From the graph: $z = 6 + 10i$, $w = 1 + i$

Step2: Multiply by conjugate of $w$

To divide, multiply numerator/denominator by $\overline{w}=1-i$:
$\frac{z}{w} = \frac{(6+10i)(1-i)}{(1+i)(1-i)}$

Step3: Expand numerator and denominator

Numerator: $(6)(1) + 6(-i) + 10i(1) + 10i(-i) = 6 -6i +10i -10i^2 = 6 +4i +10 = 16 +4i$
Denominator: $1^2 - i^2 = 1 - (-1) = 2$

Step4: Simplify the fraction

$\frac{16 +4i}{2} = 8 + 2i$

Answer:

8 + 2i