1970 AMC 12 第 13 题

Question

对所有正数

a

和

b

，定义二元运算

*

为

a*b=a^b

。则对任意正数

a

、

b

、

c

、

n

，恒有：

Given the binary operation

*

defined by

a*b=a^b

for all positive numbers

a

and

b.

Then for all positive

a,

b,

c,

n,

we have:

a*b=b*a

a*(b*c)=(a*b)*c

a*(b^n)=(a*n)*b

(a*b)^n=a*(bn)

以上都不对

None of these

先根据定义替换每个

*

，再比较等式两边

Replace every

*

using the definition before comparing the two sides

对于选项 D，两边都化简成指数为

bn

的幂

For choice D, both sides simplify to a power with exponent

bn

对于选项 D，

\begin{aligned} (a*b)^n&=(a^b)^n=a^{bn},\\ a*(bn)&=a^{bn} \end{aligned}\text{。}

因此这个恒等式总是成立。其他选项会断言诸如

a^b=b^a

或

a^{b^c}=a^{bc}

之类并非恒成立的等式。

因此，正确答案是 D。

For choice D,

\begin{aligned} (a*b)^n&=(a^b)^n=a^{bn},\\ a*(bn)&=a^{bn}. \end{aligned}

Thus that identity always holds. The other choices would assert false general identities such as

a^b=b^a

or

a^{b^c}=a^{bc}.

Therefore, the correct answer is D.

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