Question

what is the simplest form of \frac{\sqrt3{81x^{10}}}{\sqrt3{3x}}?

• \\(3x\\)
• \\(3x^3\\)
• \\(3x^4\sqrt{3x}\\)
• \\(3x^3\sqrt3{x^2}\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Quotient Property of Radicals",
"Simplifying Cube Roots"
],
"new_concepts": [],
"current_concepts": [
"Quotient Property of Radicals",
"Simplifying Cube Roots",
"Exponent Rules"
]
}
</pre_analysis>

<reasoning>

Apply the quotient property of radicals

Using the Quotient Property of Radicals knowledge point
\[
\frac{\sqrt[3]{81x^{10}}}{\sqrt[3]{3x}} = \sqrt[3]{\frac{81x^{10}}{3x}}
\]

Simplify the radicand

Using the Exponent Rules knowledge point
\[
\frac{81x^{10}}{3x} = 27x^9
\]

Simplify the cube root

Using the Simplifying Cube Roots knowledge point
\[
\sqrt[3]{27x^9} = \sqrt[3]{3^3 \cdot (x^3)^3} = 3x^3
\]
</reasoning>

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

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Quotient Property of Radicals",
"Simplifying Cube Roots"
],
"new_concepts": [],
"current_concepts": [
"Quotient Property of Radicals",
"Simplifying Cube Roots",
"Exponent Rules"
]
}
</pre_analysis>

<reasoning>

Apply the quotient property of radicals

Using the Quotient Property of Radicals knowledge point
\[
\frac{\sqrt[3]{81x^{10}}}{\sqrt[3]{3x}} = \sqrt[3]{\frac{81x^{10}}{3x}}
\]

Simplify the radicand

Using the Exponent Rules knowledge point
\[
\frac{81x^{10}}{3x} = 27x^9
\]

Simplify the cube root

Using the Simplifying Cube Roots knowledge point
\[
\sqrt[3]{27x^9} = \sqrt[3]{3^3 \cdot (x^3)^3} = 3x^3
\]
</reasoning>

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

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