1990 AMC 12 第 5 题

Question

下列各数中，哪一个最大？

Which of these numbers is largest?

\sqrt{\sqrt[3]{5\cdot6}}

\sqrt{6\sqrt[3]5}

\sqrt{5\sqrt[3]6}

\sqrt[3]{5\sqrt6}

\sqrt[3]{6\sqrt5}

所有选项均为正数，因此可分别取六次方

Every choice is positive, so raise each one to the sixth power

将每个六次方写成

5^a6^b

的形式

Express each sixth power as

5^a6^b

对五个正数分别取六次方不改变它们的大小顺序，依次得到

\begin{aligned} 30,\qquad &6^3\cdot5=1080,\\ 5^3\cdot6&=750,\\ 5^2\cdot6&=150,\qquad 6^2\cdot5=180 \end{aligned}\text{。}

其中第二个最大。

所以正确答案是 B。

Raising the five positive choices to the sixth power preserves their order and gives, respectively,

\begin{aligned} 30,\qquad &6^3\cdot5=1080,\\ 5^3\cdot6&=750,\\ 5^2\cdot6&=150,\qquad 6^2\cdot5=180. \end{aligned}

The largest is the second.

Thus the correct answer is B.

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