1958 AHSME Problems/Problem 8

Problem

Which of these four numbers $\sqrt{\pi^2},\,\sqrt[3]{.8},\,\sqrt[4]{.00016},\,\sqrt[3]{-1}\cdot \sqrt{(.09)^{-1}}$ , is (are) rational:

$\textbf{(A)}\ \text{none}\qquad \textbf{(B)}\ \text{all}\qquad \textbf{(C)}\ \text{the first and fourth}\qquad \textbf{(D)}\ \text{only the fourth}\qquad \textbf{(E)}\ \text{only the first}$

Solution

$\sqrt{\pi^2}=\pi$ is not rational, so eliminate choices $(B)$ , $(C)$ , and $(E)$ . Note that $(-1)^3 = (-1)$ , so $\sqrt[3]{-1}=-1$ is rational. We have found one rational number, so we can eliminate choice $(A)$ . The answer is $\boxed{ \text{(D)}}$ .

1958 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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1958 AHSME Problems/Problem 8

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