Question

multiply.

\\((2 + 4i)(-8 - 5i)\\)

\\((2 + 4i)(-8 - 5i) = \square\\)
(simplify your answer. type your answer in the form \\(a + bi\\).)

Explanation:

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

Step 1: Expand the expression

To multiply two complex numbers, we use the distributive property (often called the FOIL method) just like we do with binomials:

$$ (2 + 4i)(-8 - 5i) = 2(-8) + 2(-5i) + 4i(-8) + 4i(-5i) $$

Multiply each term:

$$ = -16 - 10i - 32i - 20i^2 $$

Step 2: Simplify using \( i^2 = -1 \)

The imaginary unit \( i \) has the special property that \( i^2 = -1 \). We substitute this value into our expression:

$$ = -16 - 10i - 32i - 20(-1) $$

$$ = -16 - 10i - 32i + 20 $$

Step 3: Combine like terms

Group the real parts together and the imaginary parts together:

• Real parts: \( -16 + 20 = 4 \)
• Imaginary parts: \( -10i - 32i = -42i \)

Combine them into the standard form \( a + bi \):

$$ 4 - 42i $$

Answer:

$$ 4 - 42i $$