6-(m-n)6-12mn(m2-n2)2.jpeg "(m+n)6-(m-n)6-12mn(m2-n2)2")
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.