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

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Question

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

- 1
- 3
- 12
- no solution

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Exponential Equations", "One-to-One Property" ], "new_concepts": [], "current_concepts": [ "Exponential Equations", "One-to-One Property" ] } ## 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.

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

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QUESTION IMAGE

Question

Explanation:

Express bases with a common base

Substitute and apply exponent rules

Equate exponents and solve

Answer: