2009 AMC 10A Problem 13 — Solution | LIVE by Po-Shen Loh

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2009 AMC 10A Problem 13

Below is the professionally curated solution for Problem 13 of the 2009 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 10A solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:exponent prime factorization

Difficulty rating: 1280

13.

Suppose that $P = 2^m$ and $Q = 3^n.$ Which of the following is equal to $12^{mn}$ for every pair of integers $(m, n)?$

$P^2 Q$

$P^n Q^m$

$P^n Q^{2m}$

$P^{2m} Q^n$

$P^{2n} Q^m$

Solution:

Since $12 = 2^2 \cdot 3,$ $12^{mn} = 2^{2mn} \cdot 3^{mn} = (2^m)^{2n} \cdot (3^n)^m = P^{2n} Q^m.$

Thus, the correct answer is E.

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