When a=3 , b=0 , c=-2 , find the value of : ` 1/2a^4 + 2/3b^2c^2 - 1/9a^2c^2 +c^3` .

Answer

To find the value of the expression \( \frac{1}{2}a^4 + \frac{2}{3}b^2c^2 - \frac{1}{9}a^2c^2 + c^3 \) when \( a = 3 \), \( b = 0 \), and \( c = -2 \), we will substitute the values of \( a \), \( b \), and \( c \) step by step and simplify the expression.

### Step-by-Step Solution:

1. **Write down the expression**:
\[ \frac{1}{2}a^4 + \frac{2}{3}b^2c^2 - \frac{1}{9}a^2c^2 + c^3 \]

2. **Substitute the values of \( a \), \( b \), and \( c \)**:
\[ \frac{1}{2}(3)^4 + \frac{2}{3}(0)^2(-2)^2 - \frac{1}{9}(3)^2(-2)^2 + (-2)^3 \]

3. **Calculate \( a^4 \)**:
\[ (3)^4 = 81 \]
Thus, the first term becomes:
\[ \frac{1}{2} \times 81 = \frac{81}{2} \]

4. **Calculate \( b^2c^2 \)**:
\[ (0)^2 = 0 \quad \text{and} \quad (-2)^2 = 4 \]
Therefore, the second term becomes:
\[ \frac{2}{3} \times 0 \times 4 = 0 \]

5. **Calculate \( a^2c^2 \)**:
\[ (3)^2 = 9 \quad \text{and} \quad (-2)^2 = 4 \]
Thus, the third term becomes:
\[ -\frac{1}{9} \times 9 \times 4 = -\frac{36}{9} = -4 \]

6. **Calculate \( c^3 \)**:
\[ (-2)^3 = -8 \]

7. **Combine all the terms**:
\[ \frac{81}{2} + 0 - 4 - 8 \]
Simplifying this gives:
\[ \frac{81}{2} - 4 - 8 = \frac{81}{2} - \frac{8}{2} - \frac{16}{2} = \frac{81 - 8 - 16}{2} = \frac{57}{2} \]

### Final Answer:
\[ \frac{57}{2} \]