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(3-2i)(5+4i)

less than a minute read Jun 16, 2024
(3-2i)(5+4i)

Multiplying Complex Numbers: (3-2i)(5+4i)

This article will guide you through the process of multiplying the complex numbers (3-2i) and (5+4i).

Understanding Complex Numbers

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

  • a and b are real numbers
  • 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 (or FOIL method):

(3-2i)(5+4i) = (3)(5) + (3)(4i) + (-2i)(5) + (-2i)(4i)

Simplifying the Expression

  1. Multiply the terms:

    • (3)(5) = 15
    • (3)(4i) = 12i
    • (-2i)(5) = -10i
    • (-2i)(4i) = -8i²

  2. Substitute i² with -1:

    • -8i² = -8(-1) = 8

  3. Combine the real and imaginary terms:

    • 15 + 8 + 12i - 10i

  4. Simplify:

    • 23 + 2i

Conclusion

Therefore, the product of (3-2i) and (5+4i) is 23 + 2i.

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