(8-3i)(3+2i)

Jasonbradley

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Jun 16, 2024

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Multiplying Complex Numbers: (8-3i)(3+2i)

This article will guide you through the process of multiplying two complex numbers: (8-3i) and (3+2i).

Understanding Complex Numbers

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1 (i² = -1).

Multiplication Process

To multiply complex numbers, we use the distributive property, just like with regular binomials:

1. FOIL (First, Outer, Inner, Last): We multiply each term in the first complex number by each term in the second complex number.

   • First: 8 * 3 = 24
   • Outer: 8 * 2i = 16i
   • Inner: -3i * 3 = -9i
   • Last: -3i * 2i = -6i²

2. Simplify: Combine the real terms and the imaginary terms. Remember that i² = -1.

   • 24 + 16i - 9i + 6 = 30 + 7i

Result

Therefore, the product of (8-3i) and (3+2i) is 30 + 7i.

Key Takeaways

• Complex number multiplication involves the distributive property and the simplification of i².
• The result of multiplying two complex numbers is another complex number.
• Always combine real terms and imaginary terms separately.