Simplify:
`(4x+7)^(3)+(4x-7)^(3)`

Answer

To simplify the expression \((4x+7)^{3} + (4x-7)^{3}\), we can use the formula for the sum of cubes. The sum of cubes can be expressed as:

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

In our case, we can let:

- \(a = 4x + 7\)
- \(b = 4x - 7\)

Now, we can find \(a + b\) and \(a^2 - ab + b^2\).

### Step 1: Calculate \(a + b\)

\[ a + b = (4x + 7) + (4x - 7) = 4x + 7 + 4x - 7 = 8x \]

### Step 2: Calculate \(a^2\)

\[ a^2 = (4x + 7)^2 = (4x)^2 + 2 \cdot (4x) \cdot 7 + 7^2 = 16x^2 + 56x + 49 \]

### Step 3: Calculate \(b^2\)

\[ b^2 = (4x - 7)^2 = (4x)^2 - 2 \cdot (4x) \cdot 7 + 7^2 = 16x^2 - 56x + 49 \]

### Step 4: Calculate \(ab\)

\[ ab = (4x + 7)(4x - 7) = (4x)^2 - 7^2 = 16x^2 - 49 \]

### Step 5: Calculate \(a^2 - ab + b^2\)

Now we can substitute \(a^2\), \(b^2\), and \(ab\) into the expression \(a^2 - ab + b^2\):

\[ a^2 - ab + b^2 = (16x^2 + 56x + 49) - (16x^2 - 49) + (16x^2 - 56x + 49) \]

Simplifying this step-by-step:

1. Combine like terms:
\[ = 16x^2 + 56x + 49 - 16x^2 + 49 + 16x^2 - 56x + 49 \]
2. The \(16x^2\) terms cancel out:
\[ = (56x - 56x) + (49 + 49 + 49) = 0 + 147 = 147 \]

### Step 6: Substitute back into the sum of cubes formula

Now, we can substitute \(a + b\) and \(a^2 - ab + b^2\) back into the sum of cubes formula:

\[ (4x + 7)^3 + (4x - 7)^3 = (8x)(147) = 1176x \]

### Final Answer

Thus, the simplified expression is:

\[ \boxed{1176x} \]