%5E3-(2x-3y)%5E3-simplify.jpeg "(2x+3y)^3-(2x-3y)^3 Simplify")
Simplifying the Expression: (2x+3y)^3 - (2x-3y)^3
This problem involves simplifying a difference of cubes expression. To do this, we can use the following algebraic identity:
a³ - b³ = (a - b)(a² + ab + b²)
Let's apply this identity to our given expression:
1. Identify a and b
In our case:
- a = (2x + 3y)
- b = (2x - 3y)
2. Substitute into the identity
(2x+3y)³ - (2x-3y)³ = [(2x + 3y) - (2x - 3y)][(2x + 3y)² + (2x + 3y)(2x - 3y) + (2x - 3y)²]
3. Simplify the expression
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Simplify the first factor: (2x + 3y) - (2x - 3y) = 2x + 3y - 2x + 3y = 6y
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Simplify the second factor:
- (2x + 3y)² = 4x² + 12xy + 9y²
- (2x + 3y)(2x - 3y) = 4x² - 9y²
- (2x - 3y)² = 4x² - 12xy + 9y²
Therefore, the second factor becomes: 4x² + 12xy + 9y² + 4x² - 9y² + 4x² - 12xy + 9y² = 12x² + 9y²
4. Combine the simplified factors
(2x + 3y)³ - (2x - 3y)³ = (6y)(12x² + 9y²)
5. Further simplification
(2x + 3y)³ - (2x - 3y)³ = 72x²y + 54y³
Therefore, the simplified form of the expression (2x + 3y)³ - (2x - 3y)³ is 72x²y + 54y³.