Question

which are perfect cubes? check all that apply.

64
\\(x^{16}\\)
\\(8x^3\\)
\\(27x^4\\)
\\(81x^6\\)
\\(125x^9\\)

Explanation:

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<reasoning>

Analyze numerical coefficients and variable exponents

\[

$$\begin{aligned} &\text{A term } a x^n \text{ is a perfect cube if } a = b^3 \text{ for some integer } b, \text{ and } n = 3k \text{ for some integer } k \ge 0. \end{aligned}$$

\]

Evaluate each option

\[

$$\begin{aligned} &64 = 4^3 \quad (\text{Perfect cube})\\ &x^{16} \implies 16 \text{ is not divisible by } 3 \quad (\text{Not a perfect cube})\\ &8x^3 = (2x)^3 \quad (\text{Perfect cube})\\ &27x^4 \implies 4 \text{ is not divisible by } 3 \quad (\text{Not a perfect cube})\\ &81x^6 \implies 81 = 3^4 \text{ is not a perfect cube} \quad (\text{Not a perfect cube})\\ &125x^9 = (5x^3)^3 \quad (\text{Perfect cube}) \end{aligned}$$

\]

Identify correct choices

\[
\text{The perfect cubes are: } 64, \ 8x^3, \text{ and } 125x^9.
\]
</reasoning>

<answer>
<mcq-correct>64</mcq-correct>
<mcq-option>\(x^{16}\)</mcq-option>
<mcq-correct>\(8x^3\)</mcq-correct>
<mcq-option>\(27x^4\)</mcq-option>
<mcq-option>\(81x^6\)</mcq-option>
<mcq-correct>\(125x^9\)</mcq-correct>
</answer>

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

Answer:

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],
"new_concepts": [],
"current_concepts": [
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</pre_analysis>

<reasoning>

Analyze numerical coefficients and variable exponents

\[

$$\begin{aligned} &\text{A term } a x^n \text{ is a perfect cube if } a = b^3 \text{ for some integer } b, \text{ and } n = 3k \text{ for some integer } k \ge 0. \end{aligned}$$

\]

Evaluate each option

\[

$$\begin{aligned} &64 = 4^3 \quad (\text{Perfect cube})\\ &x^{16} \implies 16 \text{ is not divisible by } 3 \quad (\text{Not a perfect cube})\\ &8x^3 = (2x)^3 \quad (\text{Perfect cube})\\ &27x^4 \implies 4 \text{ is not divisible by } 3 \quad (\text{Not a perfect cube})\\ &81x^6 \implies 81 = 3^4 \text{ is not a perfect cube} \quad (\text{Not a perfect cube})\\ &125x^9 = (5x^3)^3 \quad (\text{Perfect cube}) \end{aligned}$$

\]

Identify correct choices

\[
\text{The perfect cubes are: } 64, \ 8x^3, \text{ and } 125x^9.
\]
</reasoning>

<answer>
<mcq-correct>64</mcq-correct>
<mcq-option>\(x^{16}\)</mcq-option>
<mcq-correct>\(8x^3\)</mcq-correct>
<mcq-option>\(27x^4\)</mcq-option>
<mcq-option>\(81x^6\)</mcq-option>
<mcq-correct>\(125x^9\)</mcq-correct>
</answer>

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