## Simplifying the Expression (a + b)³ - (a - b)³

This article will explore the simplification of the expression **(a + b)³ - (a - b)³**. We'll break down the process step-by-step and explore the underlying concepts.

### Understanding the Problem

The expression involves the cube of two binomial terms, where the only difference is the sign connecting 'a' and 'b'. This pattern suggests a potential simplification using algebraic identities.

### Applying Algebraic Identities

We can utilize the following identities to simplify the expression:

**(x + y)³ = x³ + 3x²y + 3xy² + y³****(x - y)³ = x³ - 3x²y + 3xy² - y³**

Let's apply these identities to our expression:

**(a + b)³ - (a - b)³ = (a³ + 3a²b + 3ab² + b³) - (a³ - 3a²b + 3ab² - b³) **

### Simplifying the Expression

Now, we can remove the parentheses and combine like terms:

**(a³ + 3a²b + 3ab² + b³) - (a³ - 3a²b + 3ab² - b³) = a³ + 3a²b + 3ab² + b³ - a³ + 3a²b - 3ab² + b³ **

Notice that 'a³' and '-a³' cancel out, as do '3ab²' and '-3ab²'. This leaves us with:

**= 6a²b + 2b³**

### Final Result

The simplified form of the expression **(a + b)³ - (a - b)³** is **6a²b + 2b³**.

### Key Takeaways

This exercise demonstrates the power of algebraic identities in simplifying complex expressions. By recognizing patterns and applying these identities, we can achieve a more concise and manageable form. Remember, understanding these concepts will be beneficial for further mathematical explorations.