AMC12 2002 A

AMC12 2002 A · Q3

AMC12 2002 A · Q3. It mainly tests Exponents & radicals, Casework.

According to the standard convention for exponentiation, \[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\] If the order in which the exponentiations are performed is changed, how many other values are possible?

根据指数运算的标准约定， \[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\] 如果改变进行指数运算的顺序，还可能得到多少个其他的值？

(A) 0 0

(B) 1 1

(D) 3 3

(E) 4 4

Answer

Correct choice: (B)

正确答案：（B）

Solution

The best way to solve this problem is by simple brute force. It is convenient to drop the usual way how exponentiation is denoted, and to write the formula as $2\uparrow 2\uparrow 2\uparrow 2$, where $\uparrow$ denotes exponentiation. We are now examining all ways to add parentheses to this expression. There are 5 ways to do so: 1. $2\uparrow (2\uparrow (2\uparrow 2))$ 2. $2\uparrow ((2\uparrow 2)\uparrow 2)$ 3. $((2\uparrow 2)\uparrow 2)\uparrow 2$ 4. $(2\uparrow (2\uparrow 2))\uparrow 2$ 5. $(2\uparrow 2)\uparrow (2\uparrow 2)$ We can note that $2\uparrow (2\uparrow 2) = (2\uparrow 2)\uparrow 2 =16$. Therefore options 1 and 2 are equal, and options 3 and 4 are equal. Option 1 is the one given in the problem statement. Thus we only need to evaluate options 3 and 5. $((2\uparrow 2)\uparrow 2)\uparrow 2 = 16\uparrow 2 = 256$ $(2\uparrow 2)\uparrow (2\uparrow 2) = 4 \uparrow 4 = 256$ Thus the only other result is $256$, and our answer is $\boxed{\textbf{(B) } 1}$.

解决此题最好的方法是直接穷举。为方便起见，不用通常的指数记号，而把式子写成 $2\uparrow 2\uparrow 2\uparrow 2$，其中 $\uparrow$ 表示乘方。现在考察给该表达式添加括号的所有方式，共有 5 种： 1. $2\uparrow (2\uparrow (2\uparrow 2))$ 2. $2\uparrow ((2\uparrow 2)\uparrow 2)$ 3. $((2\uparrow 2)\uparrow 2)\uparrow 2$ 4. $(2\uparrow (2\uparrow 2))\uparrow 2$ 5. $(2\uparrow 2)\uparrow (2\uparrow 2)$ 注意 $2\uparrow (2\uparrow 2) = (2\uparrow 2)\uparrow 2 =16$。因此选项 1 与 2 相同，选项 3 与 4 相同。选项 1 就是题目中给出的情况，所以只需计算选项 3 和 5。 $((2\uparrow 2)\uparrow 2)\uparrow 2 = 16\uparrow 2 = 256$ $(2\uparrow 2)\uparrow (2\uparrow 2) = 4 \uparrow 4 = 256$ 因此唯一的其他结果是 $256$，答案为 $\boxed{\textbf{(B) } 1}$。

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