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(6+5i)(3-2i)

2 min read Jun 16, 2024
(6+5i)(3-2i)

Multiplying Complex Numbers: (6+5i)(3-2i)

This article will guide you through the process of multiplying two complex numbers: (6+5i) 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
  • i is the imaginary unit, defined as the square root of -1 (i² = -1)

Multiplying Complex Numbers

To multiply two complex numbers, we use the distributive property, just like we would with any binomial multiplication.

  1. Expand the product:

    (6+5i)(3-2i) = 6(3) + 6(-2i) + 5i(3) + 5i(-2i)

  2. Simplify:

    = 18 - 12i + 15i - 10i²

  3. Substitute i² = -1:

    = 18 - 12i + 15i + 10

  4. Combine real and imaginary terms:

    = (18 + 10) + (-12 + 15)i

  5. Final result:

    = 28 + 3i

Conclusion

Therefore, the product of (6+5i) and (3-2i) is 28 + 3i.

This process can be applied to multiply any pair of complex numbers, remembering to use the distributive property and substitute i² with -1.

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