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2015 Math Kangaroo

Problem 15

Problem 15 · 2015 Math Kangaroo Hard
Algebra & Patterns substitution

The geometric mean of n numbers is defined as the nth root of the product of all n numbers, that is n√x1 · x2 · … · xn. We have six numbers. The geometric mean of three of them is 3, the geometric mean of the other three is 12. How big is the geometric mean of all six numbers?

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Answer: B — 6
Show hints
Hint 1 of 2
A geometric mean of 3 numbers tells you their product.
Still stuck? Show hint 2 →
Hint 2 of 2
Multiply the two products, then take the sixth root.
Show solution
Approach: combine the two products under one sixth root
  1. Geometric mean 3 of three numbers means their product is 3³ = 27; geometric mean 12 means the other product is 12³ = 1728.
  2. All six multiply to 27×1728 = 46656 = 6⁶.
  3. The geometric mean of all six is the sixth root, 6 (B).
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