Question

simplify each expression. select the correct answer from the drop-down menu.

\\-6(3)(-2i) = \\

\\2(3-i)(-2+4i) = \\

Explanation:

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

Step 1: Analyze the first expression

The first expression is:

$$ -6(3i)(-2i) $$

Multiply the real coefficients together:

$$ -6 \cdot 3 \cdot (-2) = 36 $$

Multiply the imaginary units together:

$$ i \cdot i = i^2 $$

Combine these results:

$$ 36i^2 $$

Step 2: Simplify using the definition of \( i^2 \)

Since \( i^2 = -1 \), substitute this value into the expression:

$$ 36(-1) = -36 $$

Step 3: Analyze the second expression

The second expression is:

$$ 2(3 - i)(-2 + 4i) $$

First, expand the product of the two complex numbers in the parentheses using the FOIL method (First, Outer, Inner, Last):

$$ (3 - i)(-2 + 4i) = 3(-2) + 3(4i) + (-i)(-2) + (-i)(4i) $$

$$ = -6 + 12i + 2i - 4i^2 $$

Substitute \( i^2 = -1 \):

$$ = -6 + 14i - 4(-1) $$

$$ = -6 + 14i + 4 $$

$$ = -2 + 14i $$

Now, multiply the entire result by the outer constant \( 2 \):

$$ 2(-2 + 14i) = -4 + 28i $$

Answer:

• For the first drop-down: -36
• For the second drop-down: -4 + 28i