2009 AMC 12A Problem 6

Problem 6 of 25EasierAlgebra

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)?

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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.

Concepts: exponent · substitution

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.