Question

(3 - 4i)(5 + 6i)

Explanation:

Step1: Use the distributive property (FOIL method)

Multiply each term in the first complex number by each term in the second complex number:
\((3 - 4i)(5 + 6i)=3\times5 + 3\times6i-4i\times5-4i\times6i\)

Step2: Simplify each term

Calculate each product:
\(3\times5 = 15\), \(3\times6i = 18i\), \(-4i\times5=-20i\), \(-4i\times6i=-24i^{2}\)
So the expression becomes \(15 + 18i-20i-24i^{2}\)

Step3: Recall that \(i^{2}=-1\)

Substitute \(i^{2}=-1\) into the expression:
\(-24i^{2}=-24\times(-1) = 24\)
Now the expression is \(15 + 18i-20i + 24\)

Step4: Combine like terms

Combine the real parts and the imaginary parts:
Real parts: \(15 + 24=39\)
Imaginary parts: \(18i-20i=-2i\)
So the result is \(39-2i\)

Answer:

\(39 - 2i\)