Question

let a = 3 - 5i and b = -1 + 7i. which of the following are true statements? check all that apply.
ab = 38 + 16i
ab = 32 + 26i
\\(\frac{a}{b} = -\frac{19 - 8i}{25}\\)
ab = 2 + 2i
\\(\frac{a}{b} = \frac{21 + 5i}{7}\\)
\\(\frac{a}{b} = -\frac{19 + 8i}{25}\\)

Explanation:

Step1: Calculate product $ab$

$$\begin{align*} ab&=(3-5i)(-1+7i)\\ &=3(-1)+3(7i)-5i(-1)-5i(7i)\\ &=-3+21i+5i-35i^2\\ &=-3+26i-35(-1)\\ &=-3+26i+35\\ &=32+26i \end{align*}$$

Step2: Calculate quotient $\frac{a}{b}$

First find conjugate of $b$: $\overline{b}=-1-7i$

$$\begin{align*} \frac{a}{b}&=\frac{3-5i}{-1+7i}\cdot\frac{-1-7i}{-1-7i}\\ &=\frac{(3-5i)(-1-7i)}{(-1)^2-(7i)^2}\\ &=\frac{3(-1)+3(-7i)-5i(-1)-5i(-7i)}{1-49i^2}\\ &=\frac{-3-21i+5i+35i^2}{1-49(-1)}\\ &=\frac{-3-16i+35(-1)}{1+49}\\ &=\frac{-3-16i-35}{50}\\ &=\frac{-38-16i}{50}\\ &=-\frac{19+8i}{25} \end{align*}$$

Answer:

• $ab = 32 + 26i$
• $\frac{a}{b} = -\frac{19 + 8i}{25}$