Question

multiply each pair of factors. type the product in the space provided.

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

\\((4 - 5i)(4 + 5i) = \\)

\\((-3 + 8i)(-3 - 8i) = \\)

Explanation:

🆕 New Concept Discovered: Multiplying Complex Conjugates
Multiplying complex conjugates always results in a real number.

Step 1: Multiply the first pair

To multiply complex conjugates of the form \( (a + bi)(a - bi) \), we use the identity:

$$ (a + bi)(a - bi) = a^2 + b^2 $$

For the first expression, \( (6 + 3i)(6 - 3i) \):

• Here, \( a = 6 \) and \( b = 3 \).
• Apply the formula:

$$ 6^2 + 3^2 = 36 + 9 = 45 $$

Step 2: Multiply the second pair

For the second expression, \( (4 - 5i)(4 + 5i) \):

• Here, \( a = 4 \) and \( b = 5 \).
• Apply the formula:

$$ 4^2 + 5^2 = 16 + 25 = 41 $$

Step 3: Multiply the third pair

For the third expression, \( (-3 + 8i)(-3 - 8i) \):

• Here, \( a = -3 \) and \( b = 8 \).
• Apply the formula:

$$ (-3)^2 + 8^2 = 9 + 64 = 73 $$

Answer:

• \( (6 + 3i)(6 - 3i) = 45 \)
• \( (4 - 5i)(4 + 5i) = 41 \)
• \( (-3 + 8i)(-3 - 8i) = 73 \)