1972 AMC 12 Problem 5

Question

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}},

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

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

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

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

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

None of these

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

First compare

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

with

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

, then compare

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

with each remaining number

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.

Problem 5 in other years