Question

Find the sum and the product of the complex numbers$3-2i$and $-3+5i$.

Short answer

The sum and the product of the complex numbers are $0-7i$and $-19-9i$ respectively.

Step-by-step solution

Step 1. Given Information

Two complex numbers are given as$3-2i$and$-3-5i$.

Step 2. Concept of the sum of and product of two complex numbers

Sum of two complex numbers $a+ib$and $c+id$can be calculated as

role="math" localid="1646294259645" $(a+ib)+(c+id)=(a+c)+i(b+d)$

Product of two complex numbers $a+ib$and $c+id$can be calculated as

$(a+ib)(c+id)=ac-bd+i(bc+ad)$

Step 3. Simplification 

Add the complex numbers as shown.

$$\begin{gathered} 3-2i+\left(-3-5i\right)=(3-3)+i(-2-5) \\ =0-7i \end{gathered}$$

Multiply the complex numbers as shown.

$$\begin{gathered} \left(3-2i\right)\left(-3-5i\right)=\left(3\right)\left(-3\right)+\left(-5i\right)\left(-2i\right)+\left(3\right)\left(-5i\right)+\left(-3\right)\left(-2i\right) \\ =-9+10i^2-15i+6i \\ =-9-10-9i \\ =-19-9i \end{gathered}$$