Question

what is \\(64x^{12} - 1,000\\) written as a difference of cubes?

\\((4x)^3 - 10^3\\)
\\((8x)^3 - 100^3\\)
\\((4x^4)^3 - 10^3\\)
\\((8x^4)^3 - 100^3\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Difference of Cubes",
"Perfect Cubes",
"Cube Root of Monomials"
],
"new_concepts": [],
"current_concepts": [
"Difference of Cubes",
"Perfect Cubes",
"Cube Root of Monomials"
]
}
</pre_analysis>

<reasoning>

Identify the cube root of the first term

\[
\sqrt[3]{64x^{12}} = \sqrt[3]{64} \cdot \sqrt[3]{x^{12}} = 4x^4
\]

Identify the cube root of the second term

\[
\sqrt[3]{1000} = 10
\]

Write as a difference of cubes

\[
64x^{12} - 1000 = (4x^4)^3 - 10^3
\]
</reasoning>

<answer>
<mcq-option>(A) \((4x)^3 - 10^3\)</mcq-option>
<mcq-option>(B) \((8x)^3 - 100^3\)</mcq-option>
<mcq-correct>(C) \((4x^4)^3 - 10^3\)</mcq-correct>
<mcq-option>(D) \((8x^4)^3 - 100^3\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Difference of Cubes"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Difference of Cubes",
"Perfect Cubes",
"Cube Root of Monomials"
],
"new_concepts": [],
"current_concepts": [
"Difference of Cubes",
"Perfect Cubes",
"Cube Root of Monomials"
]
}
</pre_analysis>

<reasoning>

Identify the cube root of the first term

\[
\sqrt[3]{64x^{12}} = \sqrt[3]{64} \cdot \sqrt[3]{x^{12}} = 4x^4
\]

Identify the cube root of the second term

\[
\sqrt[3]{1000} = 10
\]

Write as a difference of cubes

\[
64x^{12} - 1000 = (4x^4)^3 - 10^3
\]
</reasoning>

<answer>
<mcq-option>(A) \((4x)^3 - 10^3\)</mcq-option>
<mcq-option>(B) \((8x)^3 - 100^3\)</mcq-option>
<mcq-correct>(C) \((4x^4)^3 - 10^3\)</mcq-correct>
<mcq-option>(D) \((8x^4)^3 - 100^3\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Difference of Cubes"
]
}
</post_analysis>