Question

what is \frac{\sqrt{12x^8}}{\sqrt{3x^2}} in simplest form, where x \ge 0?

• 2\sqrt{3}x^4
• \sqrt{15}x^5
• 2x^3
• 2x^2

Explanation:

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<reasoning>

Apply the quotient property of radicals

Using the Quotient Property of Radicals knowledge point
\[
\frac{\sqrt{12x^8}}{\sqrt{3x^2}} = \sqrt{\frac{12x^8}{3x^2}}
\]

Simplify the fraction inside the radical

Using the Exponent Rules knowledge point
\[
\frac{12x^8}{3x^2} = 4x^{8-2} = 4x^6
\]

Take the square root of the simplified expression

Using the Radical Simplification knowledge point
\[
\sqrt{4x^6} = \sqrt{4} \cdot \sqrt{x^6} = 2x^3 \quad (\text{since } x \ge 0)
\]
</reasoning>

<answer>
<mcq-option>(A) \(2\sqrt{3}x^4\)</mcq-option>
<mcq-option>(B) \(\sqrt{15}x^5\)</mcq-option>
<mcq-correct>(C) \(2x^3\)</mcq-correct>
<mcq-option>(D) \(2x^2\)</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
\[
\frac{\sqrt{12x^8}}{\sqrt{3x^2}} = \sqrt{\frac{12x^8}{3x^2}}
\]

Simplify the fraction inside the radical

Using the Exponent Rules knowledge point
\[
\frac{12x^8}{3x^2} = 4x^{8-2} = 4x^6
\]

Take the square root of the simplified expression

Using the Radical Simplification knowledge point
\[
\sqrt{4x^6} = \sqrt{4} \cdot \sqrt{x^6} = 2x^3 \quad (\text{since } x \ge 0)
\]
</reasoning>

<answer>
<mcq-option>(A) \(2\sqrt{3}x^4\)</mcq-option>
<mcq-option>(B) \(\sqrt{15}x^5\)</mcq-option>
<mcq-correct>(C) \(2x^3\)</mcq-correct>
<mcq-option>(D) \(2x^2\)</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quotient Property of Radicals"
]
}
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