The factors of `(64x^(3)-216y^(3))` are :

Options

A. `(4x-6y)(16x^(2)-36y^(2))`
B. `(4x-6y)[16x^(2)+36y^(2)+24xy]`
C. `(4x+6y)(4x-6y)`
D. none of these

Answer

To factor the expression \(64x^3 - 216y^3\), we recognize that it is a difference of cubes. The formula for factoring a difference of cubes \(a^3 - b^3\) is given by:

\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

### Step-by-step Solution:

1. **Identify \(a\) and \(b\)**:
- We can express \(64\) as \(4^3\) and \(216\) as \(6^3\). Therefore, we can set:
\[ a = 4x \quad \text{and} \quad b = 6y \]

2. **Apply the difference of cubes formula**:
- Using the formula for the difference of cubes, we have:
\[ 64x^3 - 216y^3 = (4x)^3 - (6y)^3 = (4x - 6y)((4x)^2 + (4x)(6y) + (6y)^2) \]

3. **Calculate \(a^2\), \(ab\), and \(b^2\)**:
- Calculate \(a^2\):
\[ (4x)^2 = 16x^2 \]
- Calculate \(ab\):
\[ (4x)(6y) = 24xy \]
- Calculate \(b^2\):
\[ (6y)^2 = 36y^2 \]

4. **Combine the results**:
- Now substitute back into the expression:
\[ 64x^3 - 216y^3 = (4x - 6y)(16x^2 + 24xy + 36y^2) \]

5. **Final Factored Form**:
- Thus, the factors of \(64x^3 - 216y^3\) are:
\[ (4x - 6y)(16x^2 + 24xy + 36y^2) \]

### Final Answer:
The factors of \(64x^3 - 216y^3\) are \((4x - 6y)(16x^2 + 24xy + 36y^2)\).