Question

question
what is the factored form of the expression $27m^3 - 64n^3$?
$\circ (3m - 4n)(9m^2 + 12mn + 16n^2)$
$\circ (3m + 4n)(9m^2 - 12mn + 16n^2)$
$\circ (4m - 3n)(16m^2 + 12mn + 9n^2)$
$\circ (4m + 3n)(16m^2 - 12mn + 9n^2)$

Explanation:

Step1: Recall Difference of Cubes Formula

The formula for factoring the difference of two cubes is \(a^3 - b^3=(a - b)(a^2+ab + b^2)\).

Step2: Identify \(a\) and \(b\)

For the expression \(27m^3-64n^3\), we can rewrite it as \((3m)^3-(4n)^3\). So, \(a = 3m\) and \(b = 4n\).

Step3: Apply the Formula

Substitute \(a = 3m\) and \(b = 4n\) into the difference of cubes formula:
\((3m)^3-(4n)^3=(3m - 4n)[(3m)^2+(3m)(4n)+(4n)^2]\)
Simplify the second factor: \((3m)^2 = 9m^2\), \((3m)(4n)=12mn\), \((4n)^2 = 16n^2\). So the factored form is \((3m - 4n)(9m^2 + 12mn + 16n^2)\).

Answer:

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