To factor the expression \(27m^3 + 64n^3\), we need to recognize that it is a sum of cubes. The sum of cubes can be factored using the following general formula:
[tex]\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\][/tex]
In our case, the expression \(27m^3 + 64n^3\) can be written in terms of cubes:
[tex]\[27m^3 = (3m)^3\][/tex]
[tex]\[64n^3 = (4n)^3\][/tex]
So, we can rewrite the expression like this:
[tex]\[(3m)^3 + (4n)^3\][/tex]
Now, we can apply the sum of cubes formula with \(a = 3m\) and \(b = 4n\):
[tex]\[(3m)^3 + (4n)^3 = (3m + 4n)((3m)^2 - (3m)(4n) + (4n)^2)\][/tex]
We need to perform the operations inside the parentheses to get the final factors:
1. \((3m)^2 = 9m^2\)
2. \((3m)(4n) = 12mn\)
3. \((4n)^2 = 16n^2\)
Putting these together, we have:
[tex]\[(3m)^3 + (4n)^3 = (3m + 4n)(9m^2 - 12mn + 16n^2)\][/tex]
So, the factored form of the expression \(27m^3 + 64n^3\) is:
[tex]\[(3m + 4n)(9m^2 - 12mn + 16n^2)\][/tex]
Therefore, the factors of \(27m^3 + 64n^3\) are:
[tex]\[
(3m + 4n) \quad \text{and} \quad (9m^2 - 12mn + 16n^2)
\][/tex]
