1965 AHSME Problems/Problem 11

Problem

Consider the statements: \begin{align} &I: (\sqrt{-4})(\sqrt {-16}) = \sqrt{(-4)(-16)}, \\ &II: \sqrt{(-4)(-16)} = \sqrt{64}, \\ &III: \sqrt{64} = 8. \end{align} Of these the following are incorrect.

$\textbf{(A)}\ \text{none} \qquad \textbf{(B) }\ \text{I only} \qquad \textbf{(C) }\ \text{II only} \qquad \textbf{(D) }\ \text{III only}\qquad \textbf{(E) }\ \text{I and III only}$

Solution

Statement I is incorrect, because $(\sqrt{-4})(\sqrt{-16})=(i\sqrt{4})(i\sqrt{16})=i^2(\sqrt{4})(\sqrt{16})=-\sqrt{64}$ . $\newline$ Statement II is correct, because $(-4)(-16)=64$ , so $\sqrt{(-4)(-16)}=\sqrt{64}$ . $\newline$ Statement III is correct, because $8^2=64$ and $8 \geq 0$ (so it is the principal square root of 64). $\newline$ Thus, only statement I is incorrect, so we choose answer $\fbox{B}$ .

1965 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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1965 AHSME Problems/Problem 11

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