The expression `(5x-8)^3-(3x-8)^3-6x(5x-8)(3x-8)` when simplifies gives ____

Answer

To simplify the expression \((5x-8)^3 - (3x-8)^3 - 6x(5x-8)(3x-8)\), we can use the identity for the difference of cubes, which states:

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

where \(a = 5x - 8\) and \(b = 3x - 8\).

### Step 1: Identify \(a\) and \(b\)
Let:
- \(a = 5x - 8\)
- \(b = 3x - 8\)

### Step 2: Apply the difference of cubes formula
Using the difference of cubes identity, we can rewrite the expression as:

\[ (5x - 8)^3 - (3x - 8)^3 = (a - b)((5x - 8)^2 + (5x - 8)(3x - 8) + (3x - 8)^2) \]

### Step 3: Calculate \(a - b\)
Now, calculate \(a - b\):

\[ a - b = (5x - 8) - (3x - 8) = 5x - 8 - 3x + 8 = 2x \]

### Step 4: Calculate \((5x - 8)^2\), \((3x - 8)^2\), and \((5x - 8)(3x - 8)\)
Now we need to calculate the other terms:

1. \((5x - 8)^2 = 25x^2 - 80x + 64\)
2. \((3x - 8)^2 = 9x^2 - 48x + 64\)
3. \((5x - 8)(3x - 8) = 15x^2 - 40x - 24x + 64 = 15x^2 - 64x + 64\)

### Step 5: Combine the terms
Now, substitute these values back into the expression:

\[ (5x - 8)^2 + (3x - 8)^2 + (5x - 8)(3x - 8) = (25x^2 - 80x + 64) + (9x^2 - 48x + 64) + (15x^2 - 64x + 64) \]

Combine like terms:

\[ = (25x^2 + 9x^2 + 15x^2) + (-80x - 48x - 64x) + (64 + 64 + 64) \]
\[ = 49x^2 - 192x + 192 \]

### Step 6: Substitute back into the expression
Now, substitute back into the expression we derived from the difference of cubes:

\[ (5x - 8)^3 - (3x - 8)^3 = (2x)(49x^2 - 192x + 192) \]

### Step 7: Expand the expression
Now we need to expand this:

\[ = 2x(49x^2 - 192x + 192) = 98x^3 - 384x^2 + 384x \]

### Step 8: Combine with the remaining term
Now we need to subtract \(6x(5x - 8)(3x - 8)\):

Calculating \(6x(5x - 8)(3x - 8)\):

\[ = 6x(15x^2 - 64x + 64) = 90x^3 - 384x^2 + 384x \]

### Final Step: Combine everything
Now we combine:

\[ (98x^3 - 384x^2 + 384x) - (90x^3 - 384x^2 + 384x) \]

This simplifies to:

\[ = (98x^3 - 90x^3) + (-384x^2 + 384x^2) + (384x - 384x) = 8x^3 \]

Thus, the simplified expression is:

\[ \boxed{8x^3} \]