## Simplifying Complex Numbers: (9 + 5i)(9 - 5i)

This article explores the simplification of the complex number expression **(9 + 5i)(9 - 5i)** into its standard form (a + bi).

### Understanding Complex Numbers

A complex number is a number 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

The expression (9 + 5i)(9 - 5i) represents the product of two complex numbers. To simplify this expression, we can use the **difference of squares** pattern:

(a + b)(a - b) = a² - b²

Applying this pattern to our complex numbers:

(9 + 5i)(9 - 5i) = 9² - (5i)²

Now, we can simplify further by remembering that i² = -1:

9² - (5i)² = 81 - 25(-1) = 81 + 25

Finally, combining the real terms, we get:

81 + 25 = **106**

### Conclusion

Therefore, the simplified standard form of the expression (9 + 5i)(9 - 5i) is **106**. This demonstrates that the product of a complex number and its conjugate (the same number with the opposite sign on the imaginary part) results in a real number.