what equation is equivalent to $9^{x - 3} = 729$?

$9^{x - 3} = 9^{81}$
$9^{x - 3} = 9^3$
$3^{x - 3} = 3^6$
$3^{2x - 3} = 3^6$

Question

what equation is equivalent to $9^{x - 3} = 729$?

$9^{x - 3} = 9^{81}$
$9^{x - 3} = 9^3$
$3^{x - 3} = 3^6$
$3^{2x - 3} = 3^6$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Solving Exponential Equations Algebraically" ], "new_concepts": [], "current_concepts": [ "Solving Exponential Equations Algebraically" ] } ## Express both sides with a common base Using the *Solving Exponential Equations Algebraically* knowledge point $$ \begin{aligned} &9 = 3^2 \ &729 = 9^3 = (3^2)^3 = 3^6 \end{aligned} $$ ## Rewrite the original equation Using the *Solving Exponential Equations Algebraically* knowledge point $$ \begin{aligned} &9^{x - 3} = 729 \ &(3^2)^{x - 3} = 3^6 \end{aligned} $$ ## Apply exponent rules to simplify the left side Using the *Solving Exponential Equations Algebraically* knowledge point $$ \begin{aligned} &3^{2(x - 3)} = 3^6 \ &3^{2x - 6} = 3^6 \end{aligned} $$ ## Evaluate the given options Using the *Solving Exponential Equations Algebraically* knowledge point $$ \begin{aligned} &\text{Option 1: } 9^{x - 3} = 9^{81} \quad (\text{Incorrect, } 9^3 = 729 e 9^{81}) \ &\text{Option 2: } 9^{x - 3} = 9^3 \quad (\text{Correct, since } 9^3 = 729) \ &\text{Option 3: } 3^{x - 3} = 3^6 \quad (\text{Incorrect, base conversion requires } 3^{2x - 6}) \ &\text{Option 4: } 3^{2x - 3} = 3^6 \quad (\text{Incorrect, exponent should be } 2x - 6) \end{aligned} $$

(A) $9^{x - 3} = 9^{81}$

(B) $9^{x - 3} = 9^3$

(C) $3^{x - 3} = 3^6$

(D) $3^{2x - 3} = 3^6$

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

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QUESTION IMAGE

Question

Explanation:

Express both sides with a common base

Rewrite the original equation

Apply exponent rules to simplify the left side

Evaluate the given options

Answer: