If a=b= 333, c= 334 find `a^(3) + b^(3) + c^(3)- 3abc`

Answer

To solve the problem \( a^3 + b^3 + c^3 - 3abc \) where \( a = b = 333 \) and \( c = 334 \), we can use the identity:

\[ a^3 + b^3 + c^3 - 3abc = \frac{1}{2} \left( (a+b+c)(a-b)^2 + (b-c)^2 + (c-a)^2 \right) \]

### Step-by-step Solution:

1. **Identify the values:**
\[ a = 333, \quad b = 333, \quad c = 334 \]

2. **Calculate \( a + b + c \):**
\[ a + b + c = 333 + 333 + 334 = 1000 \]

3. **Calculate \( a - b \), \( b - c \), and \( c - a \):**
\[ a - b = 333 - 333 = 0 \]
\[ b - c = 333 - 334 = -1 \]
\[ c - a = 334 - 333 = 1 \]

4. **Calculate the squares:**
\[ (a - b)^2 = 0^2 = 0 \]
\[ (b - c)^2 = (-1)^2 = 1 \]
\[ (c - a)^2 = (1)^2 = 1 \]

5. **Substitute into the identity:**
\[ a^3 + b^3 + c^3 - 3abc = \frac{1}{2} \left( 1000 \cdot 0 + 1 + 1 \right) \]
\[ = \frac{1}{2} \left( 0 + 1 + 1 \right) = \frac{1}{2} \cdot 2 = 1 \]

6. **Final Calculation:**
\[ a^3 + b^3 + c^3 - 3abc = 1 \]

### Final Answer:
\[ a^3 + b^3 + c^3 - 3abc = 1 \]