## Simplifying the Expression (m+n)⁶ - (m-n)⁶ - 12mn(m²-n²)²

This article aims to simplify the given algebraic expression: **(m+n)⁶ - (m-n)⁶ - 12mn(m²-n²)²**. We will achieve this by leveraging the binomial theorem and factorization techniques.

### Applying the Binomial Theorem

Let's begin by expanding the first two terms using the binomial theorem:

**(m+n)⁶:**This can be expanded as: m⁶ + 6m⁵n + 15m⁴n² + 20m³n³ + 15m²n⁴ + 6mn⁵ + n⁶**(m-n)⁶:**Similarly, this expands to: m⁶ - 6m⁵n + 15m⁴n² - 20m³n³ + 15m²n⁴ - 6mn⁵ + n⁶

### Combining and Simplifying

Now, let's subtract the second expanded term from the first:

(m⁶ + 6m⁵n + 15m⁴n² + 20m³n³ + 15m²n⁴ + 6mn⁵ + n⁶) - (m⁶ - 6m⁵n + 15m⁴n² - 20m³n³ + 15m²n⁴ - 6mn⁵ + n⁶)

Notice that most of the terms cancel out, leaving us with:

12m⁵n + 40m³n³ + 12mn⁵

### Addressing the Third Term

Let's examine the third term: **-12mn(m²-n²)²**

We can rewrite this by expanding the squared term:

-12mn(m⁴ - 2m²n² + n⁴)

This simplifies to:

-12m⁵n + 24m³n³ - 12mn⁵

### Final Simplification

Finally, let's combine the simplified results from both parts:

(12m⁵n + 40m³n³ + 12mn⁵) + (-12m⁵n + 24m³n³ - 12mn⁵)

The terms with m⁵n and mn⁵ cancel out, leaving us with:

**64m³n³**

### Conclusion

Therefore, the simplified form of the expression (m+n)⁶ - (m-n)⁶ - 12mn(m²-n²)² is **64m³n³**. This result highlights the power of algebraic manipulations and the importance of recognizing patterns within expressions.