## Simplifying the Expression (x⁴ - y⁴) / (x - y)

This article will explore the simplification of the algebraic expression (x⁴ - y⁴) / (x - y). We will use a combination of algebraic identities and factoring techniques to arrive at a concise and simplified form.

### Understanding the Expression

The expression (x⁴ - y⁴) / (x - y) represents a fraction where the numerator is the difference of two fourth powers and the denominator is the difference of two variables. Our goal is to simplify this expression to a form that is easier to work with and understand.

### Applying the Difference of Squares Identity

We can begin simplifying the expression by recognizing that the numerator (x⁴ - y⁴) can be factored using the **difference of squares identity**:

**a² - b² = (a + b)(a - b)**

Applying this to our numerator:

x⁴ - y⁴ = (x²)² - (y²)² = (x² + y²)(x² - y²)

Now our expression becomes:

(x² + y²)(x² - y²) / (x - y)

### Factoring Further

Notice that the term (x² - y²) in the numerator is again a difference of squares. Applying the same identity again:

(x² - y²) = (x + y)(x - y)

Substituting this back into our expression:

(x² + y²)(x + y)(x - y) / (x - y)

### Simplifying the Expression

Finally, we can cancel out the common factor of (x - y) in the numerator and denominator:

(x² + y²)(x + y)

### Result

The simplified form of the expression (x⁴ - y⁴) / (x - y) is **(x² + y²)(x + y)**.

This simplification demonstrates the power of recognizing and applying algebraic identities in simplifying complex expressions.