AMC10 2009 A

AMC10 2009 A · Q13

AMC10 2009 A · Q13. It mainly tests Exponents & radicals.

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)?

设P = $2^m$且Q = $3^n$。以下哪个等价于$12^{mn}$，对任意整数对(m, n)成立？

(A) P^2 Q P^2 Q

(B) P^n Q^m P^n Q^m

(D) P^{2m} Q^n P^{2m} Q^n

(E) P^{2n} Q^m P^{2n} Q^m

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): Note that $12^{mn} = (2^2 \cdot 3)^{mn} = 2^{2mn} \cdot 3^{mn} = (2^m)^{2n} \cdot (3^n)^m = P^{2n}Q^m.$ Remark: The pair of integers $(2,1)$ shows that the other choices are not possible.

答案（E）：注意到 $12^{mn} = (2^2 \cdot 3)^{mn} = 2^{2mn} \cdot 3^{mn} = (2^m)^{2n} \cdot (3^n)^m = P^{2n}Q^m.$ 备注：整数对$(2,1)$表明其他选项是不可能的。

Topics

AL.008 Exponents & radicals