Question

multiply.

\\(2 + 8i)(5 - 20i)\\

\\(2 + 8i)(5 - 20i) = \\
(simplify your answer. type your answer

Explanation:

🆕 New Concept Discovered: Multiplying Complex Numbers
Working with the imaginary unit \( i \) where \( i^2 = -1 \)

Step 1: Distribute the terms (FOIL method)

To multiply two complex numbers, we distribute each term in the first parentheses by each term in the second parentheses, just like multiplying binomials:

$$ (2 + 8i)(5 - 20i) = (2)(5) + (2)(-20i) + (8i)(5) + (8i)(-20i) $$

Multiply the individual terms:

$$ = 10 - 40i + 40i - 160i^2 $$

Step 2: Combine like terms

Notice that the middle imaginary terms, \( -40i \) and \( 40i \), are opposites and cancel each other out:

$$ 10 + 0i - 160i^2 = 10 - 160i^2 $$

Step 3: Substitute \( i^2 = -1 \) and simplify

The imaginary unit \( i \) is defined such that its square is \( -1 \) (\( i^2 = -1 \)). Replace \( i^2 \) with \( -1 \) to simplify the expression to a real number:

$$ 10 - 160(-1) $$

$$ = 10 + 160 $$

$$ = 170 $$

Answer:

170