Question

multiply and simplify the following complex numbers: \\((-3 + 2i) \cdot (1 - i)\\)

Explanation:

Step1: Use the distributive property (FOIL method)

We multiply each term in the first complex number by each term in the second complex number:
\((-3 + 2i) \cdot (1 - i)=-3\times1+(-3)\times(-i)+2i\times1 + 2i\times(-i)\)

Step2: Simplify each term

Simplify the products: \(-3\times1=-3\), \((-3)\times(-i) = 3i\), \(2i\times1=2i\), \(2i\times(-i)=-2i^{2}\)
So the expression becomes \(-3 + 3i+2i-2i^{2}\)

Step3: Combine like terms and use \(i^{2}=- 1\)

Combine the imaginary terms: \(3i + 2i=5i\)
And substitute \(i^{2}=-1\) into \(-2i^{2}\): \(-2i^{2}=-2\times(-1) = 2\)
Now we have \(-3+5i + 2\)
Combine the real terms: \(-3 + 2=-1\)
So the simplified form is \(-1 + 5i\)

Answer:

\(-1 + 5i\)