$-\frac{\sqrt{\frac{1}{3}}}{\frac{1}{3}}, \quad -\frac{\frac{1}{5}}{\sqrt{\frac{1}{5}}}, \quad-\frac{\frac{1}{3}}{\sqrt{\frac{1}{3}}}, \quad -\frac{\sqrt{\frac{1}{5}}}{\frac{1}{5}},\quad -1$
If the 5 numbers listed above are denoted $x_1, x_2, x_3, x_4,$ and $x_5$ , so that $x_1 < x_2 < x_3 < x_4 < x_5,$ what is the value of $x_4$ ?

Correct answer marked below

-\frac{\sqrt{\frac{1}{3}}}{\frac{1}{3}}

-\frac{\frac{1}{5}}{\sqrt{\frac{1}{5}}}

-\frac{\frac{1}{3}}{\sqrt{\frac{1}{3}}}

Correct

-\frac{\sqrt{\frac{1}{5}}}{\frac{1}{5}}

-1

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Explanation

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Official Explanation

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Simplify each expression:

-\frac{\sqrt{\frac{1}{3}}}{\frac{1}{3}} = -\frac{1}{\sqrt{\frac{1}{3}}} = -\sqrt{3}

-\frac{\frac{1}{5}}{\sqrt{\frac{1}{5}}} = -\sqrt{\frac{1}{5}}

-\frac{\frac{1}{3}}{\sqrt{\frac{1}{3}}} = -\sqrt{\frac{1}{3}}

-\frac{\sqrt{\frac{1}{5}}}{\frac{1}{5}} = -\frac{1}{\sqrt{\frac{1}{5}}} = -\sqrt{5}

And the last term is

-1

.

Now compare these negative numbers (note: more negative means smaller):…

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What this question tests

Your handling of square and higher roots, including simplifying radicals.

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.