## Simplifying Complex Fractions: A Step-by-Step Guide

In mathematics, complex numbers often arise in various calculations. One common operation is dividing complex numbers. This guide will demonstrate how to simplify the complex fraction (5 - 2i) / (3 + 3i).

### 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. The **real part** is represented by **a**, and the **imaginary part** is represented by **b**.

### Simplifying the Fraction

To simplify the complex fraction (5 - 2i) / (3 + 3i), we need to eliminate the imaginary unit from the denominator. This is achieved by multiplying both the numerator and denominator by the complex conjugate of the denominator.

**1. Finding the Complex Conjugate**

The complex conjugate of a complex number is found by changing the sign of the imaginary part. The complex conjugate of (3 + 3i) is **(3 - 3i)**.

**2. Multiplying by the Complex Conjugate**

Multiply both the numerator and denominator by (3 - 3i):

```
(5 - 2i) / (3 + 3i) * (3 - 3i) / (3 - 3i)
```

**3. Expanding the Multiplication**

Expanding the multiplication in the numerator and denominator, we get:

```
(15 - 15i - 6i + 6i^2) / (9 - 9i + 9i - 9i^2)
```

**4. Simplifying Using i^2 = -1**

Remember that i^2 = -1. Substitute this into the expression:

```
(15 - 15i - 6i - 6) / (9 + 9)
```

**5. Combining Real and Imaginary Terms**

Combine the real and imaginary terms:

```
(9 - 21i) / 18
```

**6. Final Simplification**

Simplify the expression by dividing each term by 9:

```
(1 - 7/6i) / 2
```

Therefore, the simplified form of the complex fraction (5 - 2i) / (3 + 3i) is **(1 - 7/6i) / 2**.

### Conclusion

Simplifying complex fractions involves multiplying both the numerator and denominator by the complex conjugate of the denominator. This process eliminates the imaginary unit from the denominator, resulting in a simplified expression in the form **a + bi**.