Questions: What is the standard form of (7-5 i)(2+3 i) ? A. -11+29 i B. -29+11 i C. 29+11 i D. 11-29 i

Solution Steps

To find the standard form of the product of two complex numbers, we use the distributive property to expand the expression \((7-5i)(2+3i)\). This involves multiplying each term in the first complex number by each term in the second complex number. After expanding, we combine like terms and simplify, remembering that \(i^2 = -1\).

To find the product of the complex numbers \((7-5i)\) and \((2+3i)\), we use the distributive property: \[ (7-5i)(2+3i) = 7 \cdot 2 + 7 \cdot 3i - 5i \cdot 2 - 5i \cdot 3i \]

Calculate each term:

• \(7 \cdot 2 = 14\)
• \(7 \cdot 3i = 21i\)
• \(-5i \cdot 2 = -10i\)
• \(-5i \cdot 3i = -15i^2\)

Since \(i^2 = -1\), we have: \(-15i^2 = 15\)

Combine the real and imaginary parts: \[ 14 + 15 + (21i - 10i) = 29 + 11i \]

Final Answer