1972 AHSME Problems/Problem 5

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Problem 5

From among $2^{1/2}, 3^{1/3}, 8^{1/8}, 9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are

$\textbf{(A) } 3^{1/3},\ 2^{1/2}\quad  \textbf{(B) } 3^{1/3},\ 8^{1/8}\quad  \textbf{(C) } 3^{1/3},\ 9^{1/9}\quad  \textbf{(D) } 8^{1/8},\ 9^{1/9}\quad \\  \text{(E) None of these}$

Solution

$8^{1/8}$ and $9^{1/9}$ are obviously too small. We must then compare $3^{1/3}$ with $2^{1/2}$. Raising both to the power of $6$ gives $9$ and $8$ respectively. Our answer is therefore $\boxed{\textbf{(A) } 3^{1/3},\ 2^{1/2}}.$ ~lopkiloinm

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