Final answer:
The correct answer is Option A. The expression 27m³ - 64 is factored using the difference of cubes formula, resulting in (3m - 4)(9m² + 12m + 16), which is answer option (a).
Step-by-step explanation:
The student asked to factor the expression 27m³ - 64. To factor this expression, we notice that it is the difference of two cubes since 27m³ is the cube of 3m and 64 is the cube of 4. The difference of cubes can be factored using the formula a³ - b³ = (a - b)(a² + ab + b²). Applying this formula, we get:
- a = 3m
- b = 4
- a³ - b³ = (3m)³ - (4)³ = 27m³ - 64
- Factor using the difference of cubes formula: (3m - 4)(9m² + 12m + 16)
Final answer: the correct answer is option (a) (3m - 4)(9m² + 12m + 16).
To factor the expression, we can use the difference of cubes formula, which states that a³ - b³ can be factored as (a - b)(a² + ab + b²). In this case, we have (27m³ - 64) which can be written as (3m)³ - 4³.
Now, applying the difference of cubes formula, we get (3m - 4)((3m)² + (3m)(4) + 4²) which simplifies to (3m + 4)(9m² - 12m + 16).