1972 AMC 12 第 5 题

Question

在

2^{\frac{1}{2}}

、

3^{\frac{1}{3}}

、

8^{\frac{1}{8}}

、

9^{\frac{1}{9}}

中，数值最大和第二大的数依次是：

From among

2^{\frac{1}{2}},

3^{\frac{1}{3}},

8^{\frac{1}{8}},

9^{\frac{1}{9}},

those which have the greatest and the next to the greatest values, in that order, are:

3^{\frac{1}{3}}

，

2^{\frac{1}{2}}

3^{\frac{1}{3}},

2^{\frac{1}{2}}

3^{\frac{1}{3}}

，

8^{\frac{1}{8}}

3^{\frac{1}{3}},

8^{\frac{1}{8}}

3^{\frac{1}{3}}

，

9^{\frac{1}{9}}

3^{\frac{1}{3}},

9^{\frac{1}{9}}

8^{\frac{1}{8}}

，

9^{\frac{1}{9}}

8^{\frac{1}{8}},

9^{\frac{1}{9}}

以上皆非

None of these

把两个正数提升到同一个幂次来比较

Compare two roots by raising both positive numbers to a common power

先比较

3^{\frac{1}{3}}

与

2^{\frac{1}{2}}

，再把

2^{\frac{1}{2}}

与其余各数比较

First compare

3^{\frac{1}{3}}

with

2^{\frac{1}{2}}

, then compare

2^{\frac{1}{2}}

with each remaining number

取便于比较的相同幂次，可得

(3^{\frac{1}{3}})^6=9>8=(2^{\frac{1}{2}})^6\text{，}

(2^{\frac{1}{2}})^8=16>8=(8^{\frac{1}{8}})^8\text{，}

以及

(2^{\frac{1}{2}})^{18}=512>81=(9^{\frac{1}{9}})^{18}\text{。}

因此，

3^{\frac{1}{3}}

最大，

2^{\frac{1}{2}}

第二大。

因此，正确答案是 A。

Raising to convenient common powers gives

(3^{\frac{1}{3}})^6=9>8=(2^{\frac{1}{2}})^6,

(2^{\frac{1}{2}})^8=16>8=(8^{\frac{1}{8}})^8,

and

(2^{\frac{1}{2}})^{18}=512>81=(9^{\frac{1}{9}})^{18}.

Thus

3^{\frac{1}{3}}

is greatest and

2^{\frac{1}{2}}

is next.

Therefore, the correct answer is A.

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