Simplify `(4a+2b)^(3)+(4a-2b)^(3)`.

Answer

To simplify the expression \((4a + 2b)^3 + (4a - 2b)^3\), we can use the formulas for the sum of cubes and the difference of cubes.

### Step-by-Step Solution:

1. **Identify the Terms**:
Let \( A = 4a \) and \( B = 2b \). Thus, we can rewrite the expression as:
\[ (A + B)^3 + (A - B)^3 \]

2. **Use the Formulas**:
We will use the formulas:
\[ (A + B)^3 = A^3 + B^3 + 3AB(A + B) \]
\[ (A - B)^3 = A^3 - B^3 - 3AB(A - B) \]

3. **Calculate Each Part**:
- For \((A + B)^3\):
\[ (4a + 2b)^3 = (4a)^3 + (2b)^3 + 3(4a)(2b)(4a + 2b) \]
\[ = 64a^3 + 8b^3 + 3 \cdot 8ab(4a + 2b) \]
\[ = 64a^3 + 8b^3 + 24ab(4a + 2b) \]

- For \((A - B)^3\):
\[ (4a - 2b)^3 = (4a)^3 - (2b)^3 - 3(4a)(2b)(4a - 2b) \]
\[ = 64a^3 - 8b^3 - 3 \cdot 8ab(4a - 2b) \]
\[ = 64a^3 - 8b^3 - 24ab(4a - 2b) \]

4. **Combine the Results**:
Now, we add the two results:
\[ (4a + 2b)^3 + (4a - 2b)^3 = (64a^3 + 8b^3 + 24ab(4a + 2b)) + (64a^3 - 8b^3 - 24ab(4a - 2b)) \]

5. **Simplify**:
- The \(64a^3\) terms add up:
\[ 64a^3 + 64a^3 = 128a^3 \]
- The \(8b^3\) and \(-8b^3\) cancel out:
\[ 8b^3 - 8b^3 = 0 \]
- The \(24ab(4a + 2b)\) and \(-24ab(4a - 2b)\) combine:
\[ 24ab(4a + 2b) - 24ab(4a - 2b) = 24ab(4a + 2b - 4a + 2b) = 24ab(4b) \]
\[ = 96ab^2 \]

6. **Final Result**:
Combining all parts, we have:
\[ 128a^3 + 96ab^2 \]

### Final Answer:
\[ 128a^3 + 96ab^2 \]