Question

multiply and simplify the following complex numbers:
(1 + 2i)·(1 - 4i)

Explanation:

Step1: Use the distributive property (FOIL method)

Multiply each term in the first complex number by each term in the second complex number:
$$(1 + 2i) \cdot (1 - 4i)=1\times1+1\times(-4i)+2i\times1 + 2i\times(-4i)$$

Step2: Simplify each term

Simplify the products:
$$=1-4i + 2i-8i^{2}$$

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

Combine the imaginary terms and substitute \(i^{2}=-1\):
$$=1+( - 4i + 2i)-8\times(-1)$$
$$=1-2i + 8$$

Step4: Combine the real terms

Combine the real parts:
$$=(1 + 8)-2i$$
$$=9-2i$$

Answer:

\(9 - 2i\)