2009 AMC 10A Problem 13

Attempt Problem 13 of the 2009 AMC 10A 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 10A solutions, or check the answer key.

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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

Write

12 = 2^2 \cdot 3

and raise it to the power

mn

Regroup

2^{2mn} \cdot 3^{mn}

using

P = 2^m

and

Q = 3^n

Since

12 = 2^2 \cdot 3,

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

Thus, the correct answer is E.

Problem 13 in other years