AMC12 2009 A

AMC12 2009 A · Q6

AMC12 2009 A · Q6. It mainly tests Exponents & radicals, Primes & prime factorization.

Suppose that $P = 2^m$ and $Q = 3^n$. Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$?

假设 $P = 2^m$ 和 $Q = 3^n$。以下哪个表达式对每对整数 $(m, n)$ 都等于 $12^{mn}$？

(A) $P^2Q$ $P^2Q$

(B) $P^nQ^m$ $P^nQ^m$

(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

We have $12^{mn} = (2\cdot 2\cdot 3)^{mn} = 2^{2mn} \cdot 3^{mn} = (2^m)^{2n} \cdot (3^n)^m = \boxed{{E.}\ P^{2n} Q^m}$.

我们有 $12^{mn} = (2\cdot 2\cdot 3)^{mn} = 2^{2mn} \cdot 3^{mn} = (2^m)^{2n} \cdot (3^n)^m = \boxed{\bold{E)} P^{2n} Q^m}$。

Topics