Question

what is \frac{\sqrt{25x^2y^2}}{\sqrt{xy}} in simplest form? assume x \ge 0 and y \ge 0.

• 5\sqrt{xy}
• 25\sqrt{xy}
• \sqrt{5xy}
• 5xy\sqrt{xy}

Explanation:

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<reasoning>

Apply the quotient property of radicals

Using the Quotient Property of Radicals knowledge point
\[
\frac{\sqrt{25x^2y^2}}{\sqrt{xy}} = \sqrt{\frac{25x^2y^2}{xy}}
\]

Simplify the expression inside the radical

Using the Radical Simplification knowledge point
\[
\sqrt{\frac{25x^2y^2}{xy}} = \sqrt{25xy}
\]

Extract perfect squares from the radical

Using the Radical Simplification knowledge point
\[
\sqrt{25xy} = \sqrt{25} \cdot \sqrt{xy} = 5\sqrt{xy}
\]
</reasoning>

<answer>
<mcq-correct>(A) \(5\sqrt{xy}\)</mcq-correct>
<mcq-option>(B) \(25\sqrt{xy}\)</mcq-option>
<mcq-option>(C) \(\sqrt{5xy}\)</mcq-option>
<mcq-option>(D) \(5xy\sqrt{xy}\)</mcq-option>
</answer>

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Answer:

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</pre_analysis>

<reasoning>

Apply the quotient property of radicals

Using the Quotient Property of Radicals knowledge point
\[
\frac{\sqrt{25x^2y^2}}{\sqrt{xy}} = \sqrt{\frac{25x^2y^2}{xy}}
\]

Simplify the expression inside the radical

Using the Radical Simplification knowledge point
\[
\sqrt{\frac{25x^2y^2}{xy}} = \sqrt{25xy}
\]

Extract perfect squares from the radical

Using the Radical Simplification knowledge point
\[
\sqrt{25xy} = \sqrt{25} \cdot \sqrt{xy} = 5\sqrt{xy}
\]
</reasoning>

<answer>
<mcq-correct>(A) \(5\sqrt{xy}\)</mcq-correct>
<mcq-option>(B) \(25\sqrt{xy}\)</mcq-option>
<mcq-option>(C) \(\sqrt{5xy}\)</mcq-option>
<mcq-option>(D) \(5xy\sqrt{xy}\)</mcq-option>
</answer>

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