(3 + 8i)(3 – 8i) =

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Number Multiplication: (3 + 8i)(3 – 8i)

This problem involves multiplying two complex numbers. Let's break down the steps to find the solution:

Understanding Complex Numbers

Complex numbers are expressed in the form a + bi, where:

• a is the real part
• b is the imaginary part
• i is the imaginary unit, defined as the square root of -1 (i² = -1)

Multiplication of Complex Numbers

To multiply complex numbers, we use the distributive property (also known as FOIL method) just like we do with binomials:

(3 + 8i)(3 – 8i) = (3 * 3) + (3 * -8i) + (8i * 3) + (8i * -8i)

Simplifying the Expression

1. Multiply the terms: 9 - 24i + 24i - 64i²

2. Combine the imaginary terms: 9 - 64i²

3. Substitute i² with -1: 9 - 64(-1)

4. Simplify: 9 + 64

5. Final Result: 73

Conclusion

Therefore, the product of (3 + 8i) and (3 – 8i) is 73, which is a real number. This demonstrates that multiplying complex conjugates (numbers that differ only in the sign of the imaginary part) results in a real number.