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If a/b = c/d = e/f, find the value of (2*a⁴*b² + 3*a²*c² - 5*e⁴*f)/(2*b⁶ + 3*b²*d² - 5*f⁵), expressed in terms of a and b.

1. a⁴/b⁴
2. a²/b²
3. a/b
4. a³/b³

Correct answer: a⁴/b⁴

Solution

Let a/b = c/d = e/f = k. Then a = bk, c = dk, e = fk. Numerator: 2(bk)⁴ b² + 3(bk)²(dk)² - 5(fk)⁴ f = 2b⁶ k⁴ + 3 b² d² k⁴ - 5 f⁵ k⁴ = k⁴ (2b⁶ + 3 b² d² - 5 f⁵). Dividing by the denominator (2b⁶ + 3 b² d² - 5 f⁵) gives k⁴ = (a/b)⁴ = a⁴/b⁴.

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