2009 AMC 12A Problem 6 — Solution | LIVE by Po-Shen Loh

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2009 AMC 12A Problem 6

Attempt Problem 6 of the 2009 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

6.

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

$P^2 Q$

$P^n Q^m$

$P^n Q^{2m}$

$P^{2m} Q^n$

$P^{2n} Q^m$

Answer: E

Concepts:exponent substitution

Difficulty rating: 1290

Solution:

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

Thus, the correct answer is E.

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