if $32^{2c} = 8^{c+7}$, what is the value of $c$?

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Question

if $32^{2c} = 8^{c+7}$, what is the value of $c$?

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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" ], "new_concepts": [], "current_concepts": [ "Exponential Equations", "Properties of Exponents" ] } ## Express both sides with a common base of 2 $$ \begin{aligned} 32 &= 2^5 \ 8 &= 2^3 \ (2^5)^{2c} &= (2^3)^{c+7} \end{aligned} $$ ## Simplify the exponents using power rules $$ \begin{aligned} 2^{5 \cdot 2c} &= 2^{3(c+7)} \ 2^{10c} &= 2^{3c+21} \end{aligned} $$ ## Equate the exponents and solve for c $$ \begin{aligned} 10c &= 3c + 21 \ 7c &= 21 \ c &= 3 \end{aligned} $$

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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 both sides with a common base of 2

Simplify the exponents using power rules

Equate the exponents and solve for c

Answer: