(4x+2y)^3 + (4x-2y)^3

Jasonbradley

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Jun 16, 2024

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Expanding the Expression (4x + 2y)³ + (4x - 2y)³

This expression involves the sum of cubes. We can utilize the following algebraic identity to simplify it:

a³ + b³ = (a + b)(a² - ab + b²)

In our case, a = (4x + 2y) and b = (4x - 2y). Let's apply the identity:

1. Identify (a + b): (4x + 2y) + (4x - 2y) = 8x

2. Identify (a² - ab + b²):

   • (4x + 2y)² = 16x² + 16xy + 4y²
   • (4x + 2y)(4x - 2y) = 16x² - 4y²
   • (4x - 2y)² = 16x² - 16xy + 4y²
   • (a² - ab + b²) = 16x² + 16xy + 4y² - (16x² - 4y²) + 16x² - 16xy + 4y² = 32x² + 12y²

3. Substitute: (4x + 2y)³ + (4x - 2y)³ = (8x)(32x² + 12y²)

4. Simplify: (4x + 2y)³ + (4x - 2y)³ = 256x³ + 96xy²

Therefore, the simplified form of the expression (4x + 2y)³ + (4x - 2y)³ is 256x³ + 96xy².