what is the factored form of the expression (27m^3 - 64n^3)?

* ((3m - 4n)(9m^2 + 12mn + 16n^2))
* ((3m + 4n)(9m^2 - 12mn + 16n^2))
* ((4m - 3n)(16m^2 + 12mn + 9n^2))
* ((4m + 3n)(16m^2 - 12mn + 9n^2))

Question

what is the factored form of the expression (27m^3 - 64n^3)?

* ((3m - 4n)(9m^2 + 12mn + 16n^2))
* ((3m + 4n)(9m^2 - 12mn + 16n^2))
* ((4m - 3n)(16m^2 + 12mn + 9n^2))
* ((4m + 3n)(16m^2 - 12mn + 9n^2))

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Difference of Cubes", "Factoring Polynomials" ], "new_concepts": [], "current_concepts": [ "Difference of Cubes", "Factoring Polynomials" ] } ## Identify the terms as perfect cubes $$ \begin{aligned} 27m^3 &= (3m)^3 \ 64n^3 &= (4n)^3 \end{aligned} $$ ## Apply the difference of cubes formula $$ \begin{aligned} a^3 - b^3 &= (a - b)(a^2 + ab + b^2) \ \text{Let } a = 3m &\text{ and } b = 4n \end{aligned} $$ ## Substitute and simplify the factored expression $$ \begin{aligned} (3m)^3 - (4n)^3 &= (3m - 4n)((3m)^2 + (3m)(4n) + (4n)^2) \ &= (3m - 4n)(9m^2 + 12mn + 16n^2) \end{aligned} $$

(A) $(3m - 4n)(9m^2 + 12mn + 16n^2)$

(B) $(3m + 4n)(9m^2 - 12mn + 16n^2)$

(C) $(4m - 3n)(16m^2 + 12mn + 9n^2)$

(D) $(4m + 3n)(16m^2 - 12mn + 9n^2)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Difference of Cubes" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the terms as perfect cubes

Apply the difference of cubes formula

Substitute and simplify the factored expression

Answer: