2009 AMC 12A Problem 6Problem 6 of 25·Easier·AlgebraArithmeticSuppose that P=2mP = 2^mP=2m and Q=3n.Q = 3^n.Q=3n. Which of the following is equal to 12mn12^{mn}12mn for every pair of integers (m,n)?(m, n)?(m,n)?Answer choicesAP2QP^2 QP2Q1BPnQmP^n Q^mPnQm2CPnQ2mP^n Q^{2m}PnQ2m3DP2mQnP^{2m} Q^nP2mQn4EP2nQmP^{2n} Q^mP2nQm5Submit answerStuck? Show hintsShow solutionSolutionSince 12=22⋅3,12 = 2^2 \cdot 3,12=22⋅3, 12mn=22mn⋅3mn=(2m)2n(3n)m=P2nQm. \begin{aligned} 12^{mn} &= 2^{2mn} \cdot 3^{mn} \\ &= (2^m)^{2n}(3^n)^m \\ &= P^{2n}Q^m. \end{aligned} 12mn=22mn⋅3mn=(2m)2n(3n)m=P2nQm. Thus, the correct answer is E.AoPS wikiCopy problemTagged: exponent · substitution