## Squaring Complex Numbers: A Step-by-Step Guide for (6 - 8i)²

This article will explore how to square the complex number (6 - 8i). We'll break down the process into clear, manageable steps, and explain the key concepts involved.

### 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).

### Squaring a Complex Number

To square a complex number, we simply multiply it by itself.

Let's apply this to our example:

**(6 - 8i)² = (6 - 8i) * (6 - 8i)**

Now, we'll expand this product using the FOIL (First, Outer, Inner, Last) method:

**First:**6 * 6 = 36**Outer:**6 * (-8i) = -48i**Inner:**(-8i) * 6 = -48i**Last:**(-8i) * (-8i) = 64i²

Combining the terms, we get:

**36 - 48i - 48i + 64i²**

Remember that i² = -1. Substitute this into our expression:

**36 - 48i - 48i + 64(-1)**

Simplify by combining like terms:

**36 - 96i - 64**

Finally, combine the real and imaginary components:

**(36 - 64) + (-96)i**

This simplifies to:

**-28 - 96i**

### Conclusion

Therefore, (6 - 8i)² equals **-28 - 96i**. This process demonstrates how to square a complex number by applying the fundamental rules of complex arithmetic. Understanding these steps is crucial for working with complex numbers in various mathematical fields.