## Understanding the (a+b)^3-(a-b)^3 Formula

The formula **(a+b)^3-(a-b)^3 = 6ab(a+b)** is a useful algebraic identity that simplifies the expansion of the cubic expressions. It's particularly helpful for solving problems in algebra and calculus.

### Derivation of the Formula

We can derive this formula 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**

Now, subtracting the second equation from the first, 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 equation, we eliminate the terms that cancel out:

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

Finally, factoring out **6ab**, we arrive at the desired formula:

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

### Applications of the Formula

This formula has several applications, including:

**Simplifying complex algebraic expressions:**By recognizing this pattern in an expression, you can simplify it and make it easier to work with.**Solving equations:**The formula can be used to solve equations involving cubic expressions.**Calculating values:**It can be used to efficiently calculate the difference of cubes without expanding the entire expression.

### Example

Let's say we need to find the value of **(5+3)^3 - (5-3)^3**. Instead of directly expanding the cubes, we can apply the formula:

**(5+3)^3 - (5-3)^3 = 6 * 5 * 3 (5 + 3)**

Simplifying further:

**= 90 * 8**

**= 720**

Therefore, **(5+3)^3 - (5-3)^3 = 720**.

### Conclusion

The formula **(a+b)^3-(a-b)^3 = 6ab(a+b)** is a valuable tool in algebra, simplifying calculations and solving equations. Understanding its derivation and applications can be beneficial for students and anyone working with algebraic expressions.