Answer
A) adding like terms.
B) $11+7i$
C) It is a real number.
D) By multiplying by the conjugate $\frac{4+18i}{10}$
Work Step by Step
A) How do you add complex numbers?
In the addition you just add like terms. In other words, you add the real numbers with the real numbers and the imaginary with the imaginary.
B) How do you multiply $(3+5i)(2-i)$
The multiplicatiopn goes normal.
$$6-3i+10i-5(i)^{2}$$
The first thing that you need to remember is that $i$ is $\sqrt -1$ so $i^{2}$ is -1.
$$6-3i+10i+5$$ $$11+7i$$
C) Is $(3-i)(3+i)$ a real number?
You have to solve it to know the answer.
$$9+3i-3i-i^{2}$$ $$9+3i-3i+1$$ $$10$$
It is a real number.
D) how do you simplify a quotient $\frac{3+5i}{3-i}$?
You have to multiply by the conjugate.
$$(\frac{3+5i}{3-i})(\frac{3+i}{3+i})$$ $$\frac{9+3i+15i+5(i)^{2}}{9+3i-3i-i^{2}}$$ $$\frac{9+18i+5(i)^{2}}{9-i^{2}}$$ $$\frac{9+18i-5}{9+1}$$ $$\frac{4+18i}{10}$$
