(a+b)^3-(a-b)^3 Simplify

Jasonbradley

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Jun 16, 2024

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Simplifying (a + b)³ - (a - b)³

This article will guide you through simplifying the expression (a + b)³ - (a - b)³.

Understanding the Problem

The expression involves the difference of two cubes, where the cubes are (a + b)³ and (a - b)³. We can simplify this using the following algebraic identities:

• Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)
• Square of a Binomial: (a + b)² = a² + 2ab + b²
• Square of a Binomial: (a - b)² = a² - 2ab + b²

Step-by-Step Simplification

1. Apply the Difference of Cubes Identity:

   (a + b)³ - (a - b)³ = [(a + b) - (a - b)][(a + b)² + (a + b)(a - b) + (a - b)²]

2. Simplify the First Bracket:

   [(a + b) - (a - b)] = a + b - a + b = 2b

3. Expand the Second Bracket:

   [(a + b)² + (a + b)(a - b) + (a - b)²] = (a² + 2ab + b²) + (a² - b²) + (a² - 2ab + b²)

4. Combine Like Terms:

   (a² + 2ab + b²) + (a² - b²) + (a² - 2ab + b²) = 3a² + b²

5. Combine the Simplified Brackets:

   2b * (3a² + b²) = 6a²b + 2b³

Final Result

Therefore, the simplified expression of (a + b)³ - (a - b)³ is 6a²b + 2b³.

Conclusion

This simplification process demonstrates the application of algebraic identities to solve complex expressions. By recognizing the patterns and applying the appropriate identities, we can break down the expression into manageable steps and arrive at a simplified form.