## Multiplying Complex Numbers: (5 + 4i)(5 - 4i)

This article explores the multiplication of the complex numbers (5 + 4i) and (5 - 4i), illustrating the concept of **complex conjugates** and their significance.

### Understanding Complex Conjugates

The complex conjugate of a complex number is formed by simply changing the sign of the imaginary part. For example, the complex conjugate of (5 + 4i) is (5 - 4i).

Complex conjugates have a unique property: **when multiplied together, they result in a real number**. This is because the imaginary terms cancel out during multiplication.

### The Multiplication Process

Let's multiply (5 + 4i) and (5 - 4i):

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

Expanding the terms:

= 25 - 20i + 20i - 16i²

As **i² = -1**, we can substitute:

= 25 - 16(-1)

= 25 + 16

= **41**

### Conclusion

As expected, the product of the complex conjugates (5 + 4i) and (5 - 4i) is a **real number**, 41. This result highlights the key characteristic of complex conjugates and their application in simplifying complex number operations.