Question

for what value of \\(x\\) does \\(64^{3x} = 512^{2x+12}\\)?

• 1
• 3
• 12
• no solution

Explanation:

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</pre_analysis>

<reasoning>

Express bases with a common base

\[

$$\begin{aligned} 64 &= 8^2 \\ 512 &= 8^3 \end{aligned}$$

\]

Substitute and apply exponent rules

\[

$$\begin{aligned} (8^2)^{3x} &= (8^3)^{2x+12} \\ 8^{6x} &= 8^{3(2x+12)} \\ 8^{6x} &= 8^{6x+36} \end{aligned}$$

\]

Equate exponents and solve

\[

$$\begin{aligned} 6x &= 6x + 36 \\ 0 &= 36 \end{aligned}$$

\]
Since \(0 = 36\) is a contradiction, there is no solution.
</reasoning>

<answer>
<mcq-option>1</mcq-option>
<mcq-option>3</mcq-option>
<mcq-option>12</mcq-option>
<mcq-correct>no solution</mcq-correct>
</answer>

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

Answer:

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"step_cot",
"multiple_choice_formatter",
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"One-to-One Property"
],
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</pre_analysis>

<reasoning>

Express bases with a common base

\[

$$\begin{aligned} 64 &= 8^2 \\ 512 &= 8^3 \end{aligned}$$

\]

Substitute and apply exponent rules

\[

$$\begin{aligned} (8^2)^{3x} &= (8^3)^{2x+12} \\ 8^{6x} &= 8^{3(2x+12)} \\ 8^{6x} &= 8^{6x+36} \end{aligned}$$

\]

Equate exponents and solve

\[

$$\begin{aligned} 6x &= 6x + 36 \\ 0 &= 36 \end{aligned}$$

\]
Since \(0 = 36\) is a contradiction, there is no solution.
</reasoning>

<answer>
<mcq-option>1</mcq-option>
<mcq-option>3</mcq-option>
<mcq-option>12</mcq-option>
<mcq-correct>no solution</mcq-correct>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Exponential Equations"
]
}
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