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Given that 2x + 3y + 4z = 0, prove that 8x³ + 27y³ + 64z³ = 72xyz. Which identity result is correct?

1. 8x³ + 27y³ + 64z³ = 72xyz
2. 8x³ + 27y³ + 64z³ = 0
3. 8x³ + 27y³ + 64z³ = 24xyz
4. 8x³ + 27y³ + 64z³ = -72xyz

Correct answer: 8x³ + 27y³ + 64z³ = 72xyz

Solution

Set a = 2x, b = 3y, c = 4z. The condition becomes a + b + c = 0. The algebraic identity states that when a + b + c = 0, a³ + b³ + c³ = 3abc. So 8x³ + 27y³ + 64z³ = 3*(2x)(3y)(4z) = 3*24*xyz = 72xyz.

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