Question

which numbers are perfect cubes? check all that apply.

3
64
81
125
999
3,000

Explanation:

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"concepts_used": [],
"new_concepts": [
"Perfect Cubes"
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<reasoning>

Define perfect cubes

A perfect cube is an integer that can be written as the cube of another integer:
\[
n = k^3 \quad \text{for } k \in \mathbb{Z}
\]

Test each given number

Evaluate each option to see if it has an integer cube root:
\[

$$\begin{aligned} &\sqrt[3]{3} \approx 1.44 \quad (\text{Not an integer})\\ &\sqrt[3]{64} = 4 \quad (4^3 = 64)\\ &\sqrt[3]{81} \approx 4.33 \quad (\text{Not an integer})\\ &\sqrt[3]{125} = 5 \quad (5^3 = 125)\\ &\sqrt[3]{999} \approx 10.00 \quad (\text{Not an integer})\\ &\sqrt[3]{3000} \approx 14.42 \quad (\text{Not an integer}) \end{aligned}$$

\]

Identify the correct options

The numbers that are perfect cubes are \(64\) and \(125\).
</reasoning>

<answer>
<mcq-option>3</mcq-option>
<mcq-correct>64</mcq-correct>
<mcq-option>81</mcq-option>
<mcq-correct>125</mcq-correct>
<mcq-option>999</mcq-option>
<mcq-option>3,000</mcq-option>
</answer>

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

Answer:

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"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
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"concepts_used": [],
"new_concepts": [
"Perfect Cubes"
],
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"Perfect Cubes"
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</pre_analysis>

<reasoning>

Define perfect cubes

A perfect cube is an integer that can be written as the cube of another integer:
\[
n = k^3 \quad \text{for } k \in \mathbb{Z}
\]

Test each given number

Evaluate each option to see if it has an integer cube root:
\[

$$\begin{aligned} &\sqrt[3]{3} \approx 1.44 \quad (\text{Not an integer})\\ &\sqrt[3]{64} = 4 \quad (4^3 = 64)\\ &\sqrt[3]{81} \approx 4.33 \quad (\text{Not an integer})\\ &\sqrt[3]{125} = 5 \quad (5^3 = 125)\\ &\sqrt[3]{999} \approx 10.00 \quad (\text{Not an integer})\\ &\sqrt[3]{3000} \approx 14.42 \quad (\text{Not an integer}) \end{aligned}$$

\]

Identify the correct options

The numbers that are perfect cubes are \(64\) and \(125\).
</reasoning>

<answer>
<mcq-option>3</mcq-option>
<mcq-correct>64</mcq-correct>
<mcq-option>81</mcq-option>
<mcq-correct>125</mcq-correct>
<mcq-option>999</mcq-option>
<mcq-option>3,000</mcq-option>
</answer>

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