Simplify: `(3x-4y)^(3)-(3x+4y)^(3)`

Answer

To simplify the expression \((3x - 4y)^3 - (3x + 4y)^3\), we can use the identity for the difference of cubes. The identity states that:

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

In our case, let:
- \(a = 3x - 4y\)
- \(b = 3x + 4y\)

### Step 1: Identify \(a\) and \(b\)

We have:
\[ a = 3x - 4y \quad \text{and} \quad b = 3x + 4y \]

### Step 2: Calculate \(a - b\)

Now, we calculate \(a - b\):
\[ a - b = (3x - 4y) - (3x + 4y) = 3x - 4y - 3x - 4y = -8y \]

### Step 3: Calculate \(a^2\), \(ab\), and \(b^2\)

Next, we need to calculate \(a^2\), \(ab\), and \(b^2\):

1. **Calculate \(a^2\)**:
\[ a^2 = (3x - 4y)^2 = (3x)^2 - 2(3x)(4y) + (4y)^2 = 9x^2 - 24xy + 16y^2 \]

2. **Calculate \(b^2\)**:
\[ b^2 = (3x + 4y)^2 = (3x)^2 + 2(3x)(4y) + (4y)^2 = 9x^2 + 24xy + 16y^2 \]

3. **Calculate \(ab\)**:
\[ ab = (3x - 4y)(3x + 4y) = (3x)^2 - (4y)^2 = 9x^2 - 16y^2 \]

### Step 4: Substitute into the identity

Now we substitute these into the identity:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Substituting the values we calculated:
\[ = (-8y) \left((9x^2 - 24xy + 16y^2) + (9x^2 - 16y^2) + (9x^2 + 24xy + 16y^2)\right) \]

### Step 5: Simplify the expression inside the parentheses

Combine the terms:
\[ = (-8y) \left(9x^2 + 9x^2 + 9x^2 - 24xy + 24xy + 16y^2 - 16y^2\right) \]
\[ = (-8y) \left(27x^2\right) \]

### Step 6: Final simplification

Thus, we have:
\[ = -216yx^2 \]

### Final Answer
The simplified expression is:
\[ -216yx^2 \]