Multiple choice

If b is the mean proportional between a and c. Then (a – b)3 : (b – c)3 will be equal to:

  1. a3/2 : c3/2

  2. b2 : c2

  3. a2 : c2

  4. a3 : c3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given b is the mean proportional between a and c, we have b² = ac or b = √(ac). Then (a-b)³ : (b-c)³ = (a-√ac)³ : (√ac-c)³. Factoring: (a-√ac) = √a(√a-√c) and (√ac-c) = √c(√a-√c). Therefore (a-√ac)³ : (√ac-c)³ = (√a)³(√a-√c)³ : (√c)³(√a-√c)³ = a^(3/2) : c^(3/2) after cancelling (√a-√c)³. The answer a^(3/2) : c^(3/2) matches option A. Options with a²:c² or a³:c³ are incorrect as they don't match the exponent values derived.

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