## The (a+b)^3 - (a-b)^3 Identity

The identity **(a+b)^3 - (a-b)^3 = 6ab(a+b)** is a fundamental algebraic expression that simplifies the difference of two cubes. It's widely used in various fields like algebra, calculus, and physics to solve equations and simplify complex expressions. Let's break down the identity and explore its application.

### Derivation of the Identity

The identity is derived by expanding the cubes using the binomial theorem:

**(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3**
**(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3**

Subtracting the second equation from the first equation, we get:

**(a + b)^3 - (a - b)^3 = (a^3 + 3a^2b + 3ab^2 + b^3) - (a^3 - 3a^2b + 3ab^2 - b^3)**

Simplifying the expression, we obtain:

**(a + b)^3 - (a - b)^3 = 6a^2b + 2b^3 = 2b(3a^2 + b^2) = 6ab(a+b)**

### Applications of the Identity

The (a+b)^3 - (a-b)^3 identity finds its use in various applications:

**Solving equations:**The identity can be used to simplify equations involving cubes, making them easier to solve.**Factoring expressions:**The identity can be used to factor expressions of the form (a + b)^3 - (a - b)^3, which can simplify further calculations.**Simplifying complex expressions:**In more complex algebraic expressions, the identity can be applied to reduce the number of terms and make the expression more manageable.

### Example

Let's take an example to demonstrate the use of the identity:

**Simplify the expression (2x + 3)^3 - (2x - 3)^3**

Applying the identity, we get:

**(2x + 3)^3 - (2x - 3)^3 = 6(2x)(3)(2x + 3) = 36x(2x + 3)**

As you can see, the use of the identity simplified the expression and made it easier to work with.

### Conclusion

The (a+b)^3 - (a-b)^3 identity is a powerful tool in algebraic manipulations. Understanding its derivation and applications can greatly simplify complex expressions and make problem-solving easier. By recognizing the pattern of cubes in expressions, we can efficiently apply this identity to simplify expressions and solve equations more effectively.