Question

把数字 $2$ 、 $0$ 、 $2$ 和 $3$ 填入下列表达式中，每个方框放一个数字。这个表达式的最大可能值是多少？

The digits $2,$ $0,$ $2,$ and $3$ are placed in the expression below, one digit per box. What is the maximum possible value of the expression?

$0$

$8$

$9$

$16$

$18$

避免把 $0$ 放在底数位置

Avoid putting $0$ in a base

把 $0$ 用作指数，使其中一个因子变成 $1$

Use $0$ as an exponent so one factor becomes $1$

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不要把 $0$ 放在底数位置，因为那样会使整个表达式等于 $0$ 。

所以 $0$ 必须放在指数位置。指数为 $0$ 的那一项会自动等于 $1$ 。

把两个 $2$ 中的一个用于 $2^0$ ，就能把另一个 $2$ 和 $3$ 留给另一项。

另一项可以是 $2^3$ 或 $3^2$ 。因为 $3^2$ 更大，所以取 $3^2$ 。

于是最大值为 $2^0 \times 3^2 = 1 \times 9 = 9\text{。}$

所以正确答案是 C。

Note that we do not want $0$ as a base, since that would make the expression equal to $0.$

This means that $0$ must be an exponent. The number whose exponent is $0$ will automatically evaluate to $1.$

Using one of the two copies of $2$ in the factor $2^0$ leaves $2$ and $3$ for the other factor.

The other factor can then be $2^3$ or $3^2.$ Since $3^2$ is larger, we use $3^2.$

This gives us a final value of $2^0 \times 3^2 = 1 \times 9 = 9.$

Thus, C is the correct answer.