Question

multiply.

(5v^{9} \cdot 3w^{4} v^{3} \cdot 2w)

simplify your answer as much as possible.

Explanation:

<pre_analysis>
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"step_cot",
"blank_filling_formatter",
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"concepts_used": [
"Exponent Rules",
"Multiplying Monomials"
],
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</pre_analysis>

<reasoning>

Rearrange and group terms

\[
(5 \cdot 3 \cdot 2) \cdot (v^9 \cdot v^3) \cdot (w^4 \cdot w^1)
\]

Multiply coefficients and apply product rule for exponents

\[

$$\begin{aligned} 5 \cdot 3 \cdot 2 &= 30 \\ v^9 \cdot v^3 &= v^{9+3} = v^{12} \\ w^4 \cdot w^1 &= w^{4+1} = w^5 \end{aligned}$$

\]

Combine the simplified parts

\[
30v^{12}w^5
\]
</reasoning>

<answer>
Multiply.

\(5v^9 \cdot 3w^4 \cdot v^3 \cdot 2w\)

Simplify your answer as much as possible.

<blank>\(30v^{12}w^5\)</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Exponent Rules"
]
}
</post_analysis>

Answer:

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"step_cot",
"blank_filling_formatter",
"explain_then_answer",
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],
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"concepts_used": [
"Exponent Rules",
"Multiplying Monomials"
],
"new_concepts": [],
"current_concepts": [
"Exponent Rules",
"Multiplying Monomials"
]
}
</pre_analysis>

<reasoning>

Rearrange and group terms

\[
(5 \cdot 3 \cdot 2) \cdot (v^9 \cdot v^3) \cdot (w^4 \cdot w^1)
\]

Multiply coefficients and apply product rule for exponents

\[

$$\begin{aligned} 5 \cdot 3 \cdot 2 &= 30 \\ v^9 \cdot v^3 &= v^{9+3} = v^{12} \\ w^4 \cdot w^1 &= w^{4+1} = w^5 \end{aligned}$$

\]

Combine the simplified parts

\[
30v^{12}w^5
\]
</reasoning>

<answer>
Multiply.

\(5v^9 \cdot 3w^4 \cdot v^3 \cdot 2w\)

Simplify your answer as much as possible.

<blank>\(30v^{12}w^5\)</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Exponent Rules"
]
}
</post_analysis>