2009 AMC 12A 第 6 题

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所有题目均经美国数学协会（MAA）官方合法授权使用。

假设

P = 2^m

且

Q = 3^n

。对每一对整数

(m, n)

，下列哪一个都等于

12^{mn}

？

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

写成

12 = 2^2 \cdot 3

。

Write

12 = 2^2 \cdot 3

那么

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

= (2^m)^{2n}(3^n)^m

。

Then

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

= (2^m)^{2n}(3^n)^m

因为

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}\text{。}

因此，正确答案是 E。

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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