If `a ^(3) - b ^(3) - c ^(3) = 0,` then the value of `a ^(9) - b ^(9) - c ^(9) - 3a ^(3) b ^(3) c ^(3)` is :

Options

A. 1
B. 2
C. 0
D. `-1`

Answer

To solve the problem, we start with the equation given:

1. **Given Equation**:
\[ a^3 - b^3 - c^3 = 0 \]
This implies:
\[ a^3 = b^3 + c^3 \]

2. **Cubing Both Sides**:
We will cube both sides of the equation \( a^3 = b^3 + c^3 \):
\[ (a^3)^3 = (b^3 + c^3)^3 \]
This simplifies to:
\[ a^9 = (b^3 + c^3)^3 \]

3. **Expanding the Right Side**:
Using the binomial expansion:
\[ (b^3 + c^3)^3 = b^9 + c^9 + 3b^3c^3(b^3 + c^3) \]
Therefore, we have:
\[ a^9 = b^9 + c^9 + 3b^3c^3(b^3 + c^3) \]

4. **Substituting \( b^3 + c^3 \)**:
Since we know \( b^3 + c^3 = a^3 \), we can substitute:
\[ a^9 = b^9 + c^9 + 3b^3c^3a^3 \]

5. **Rearranging the Equation**:
Now, we want to find the value of:
\[ a^9 - b^9 - c^9 - 3a^3b^3c^3 \]
Substituting our expression for \( a^9 \):
\[ a^9 - b^9 - c^9 - 3a^3b^3c^3 = (b^9 + c^9 + 3b^3c^3a^3) - b^9 - c^9 - 3a^3b^3c^3 \]

6. **Simplifying**:
The terms \( b^9 \) and \( c^9 \) cancel out:
\[ = 0 \]

Thus, the final answer is:
\[ \text{Value} = 0 \]