2009 AMC 10A Problem 13

Problem 13 of 25IntermediateAlgebra Number Theory

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

Thus, the correct answer is E.

Concepts: exponent · prime factorization

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