2010 AMC 12A Problems/Problem 4

Problem

If $x<0$ , then which of the following must be positive?

$\textbf{(A)}\ \frac{x}{\left|x\right|} \qquad \textbf{(B)}\ -x^2 \qquad \textbf{(C)}\ -2^x \qquad \textbf{(D)}\ -x^{-1} \qquad \textbf{(E)}\ \sqrt[3]{x}$

Solution

$x$ is negative, so we can just place a negative value into each expression and find the one that is positive. Suppose we use $-1$ .

$\textbf{(A)} \Rightarrow \frac{-1}{|-1|} = -1$

$\textbf{(B)} \Rightarrow -(-1)^2 = -1$

$\textbf{(C)} \Rightarrow -2^{(-1)} = -\frac{1}{2}$

$\textbf{(D)} \Rightarrow -(-1)^{(-1)} = 1$

$\textbf{(E)} \Rightarrow \sqrt[3]{-1} = -1$

Obviously only $\boxed{\textbf{(D)}}$ is positive.

2010 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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All AMC 12 Problems and Solutions

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2010 AMC 12A Problems/Problem 4

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