select all the correct answers.

which expressions are equal to $\frac{6^3 \cdot 2^3}{2^3}$?

$6^3$

$2^6 \cdot 3^3$

$12^6$

$2^3 \cdot 3^3$

$12^3$

Question

select all the correct answers.

which expressions are equal to $\frac{6^3 \cdot 2^3}{2^3}$?

$6^3$

$2^6 \cdot 3^3$

$12^6$

$2^3 \cdot 3^3$

$12^3$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Exponent Rules", "Quotient of Powers" ], "new_concepts": [ "Power of a Product Rule" ], "current_concepts": [ "Exponent Rules", "Quotient of Powers", "Power of a Product Rule" ] } ## Simplify the given expression $$ \frac{6^3 \cdot 2^6}{2^3} = 6^3 \cdot 2^{6-3} = 6^3 \cdot 2^3 $$ ## Rewrite using prime factorization $$ 6^3 \cdot 2^3 = (2 \cdot 3)^3 \cdot 2^3 = 2^3 \cdot 3^3 \cdot 2^3 = 2^6 \cdot 3^3 $$ ## Evaluate the options $$ \begin{aligned} &\text{Option 1: } 6^3 \quad (\text{Incorrect})\ &\text{Option 2: } 2^6 \cdot 3^3 \quad (\text{Correct})\ &\text{Option 3: } 12^6 \quad (\text{Incorrect})\ &\text{Option 4: } 2^3 \cdot 3^3 = (2 \cdot 3)^3 = 6^3 \quad (\text{Incorrect})\ &\text{Option 5: } 12^3 = (6 \cdot 2)^3 = 6^3 \cdot 2^3 \quad (\text{Correct}) \end{aligned} $$

$\square$ $6^3$

$\square$ $2^6 \cdot 3^3$

$\square$ $12^6$

$\square$ $2^3 \cdot 3^3$

$\square$ $12^3$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Exponent Rules" ] }

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QUESTION IMAGE

Question

Explanation:

Simplify the given expression

Rewrite using prime factorization

Evaluate the options

Answer: