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(6+2i)(6-2i)-(3-4i)(3+4i)

2 min read Jun 16, 2024
(6+2i)(6-2i)-(3-4i)(3+4i)

Simplifying Complex Expressions: (6+2i)(6-2i)-(3-4i)(3+4i)

This article will demonstrate how to simplify the complex expression (6+2i)(6-2i)-(3-4i)(3+4i).

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).

Simplifying the Expression

To simplify the given expression, we will use the following properties:

  • Difference of Squares: (a + b)(a - b) = a² - b²
  • i² = -1

Let's break down the expression step-by-step:

  1. (6+2i)(6-2i): Applying the difference of squares property, we get: (6+2i)(6-2i) = 6² - (2i)² = 36 - 4i²

  2. (3-4i)(3+4i): Similarly, applying the difference of squares property: (3-4i)(3+4i) = 3² - (4i)² = 9 - 16i²

  3. Substituting i² = -1:

    • 36 - 4i² = 36 - 4(-1) = 36 + 4 = 40
    • 9 - 16i² = 9 - 16(-1) = 9 + 16 = 25

  4. Combining the results: (6+2i)(6-2i)-(3-4i)(3+4i) = 40 - 25 = 15

Conclusion

Therefore, the simplified form of the complex expression (6+2i)(6-2i)-(3-4i)(3+4i) is 15. This example showcases how utilizing the properties of complex numbers and algebraic manipulations can lead to a simplified result.

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