## Exploring Complex Number Multiplication: (6 + 3i)(6 − 3i)

This expression involves the multiplication of two complex numbers. Let's break down the steps and explore the interesting result:

### 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**).

### Multiplication Process

To multiply complex numbers, we can use the distributive property (or the FOIL method):

**(6 + 3i)(6 − 3i) = 6(6) + 6(−3i) + 3i(6) + 3i(−3i)**

### Simplifying the Expression

Let's simplify the terms:

**36 - 18i + 18i - 9i²**

Since **i² = -1**, we can substitute:

**36 - 18i + 18i + 9**

### The Result

Combining like terms, we arrive at the final result:

**(6 + 3i)(6 − 3i) = 45**

### Key Observations

- The result is a
**real number**. This is because the imaginary terms cancel each other out. - The expression (6 + 3i) and (6 − 3i) are
**conjugates**of each other. Conjugates always result in a real number when multiplied.

### Significance

This calculation demonstrates a crucial concept in complex numbers: **conjugates** provide a way to eliminate imaginary terms and obtain real number results. This property is widely used in various applications, including solving equations, simplifying expressions, and working with complex functions.