If ` a + b = 3 ` then the value of ` a^(3) + b^(3)` + 9 ab is

Options

A. 27
B. 9
C. 16
D. 81

Answer

To solve the problem, we need to find the value of \( a^3 + b^3 + 9ab \) given that \( a + b = 3 \).

### Step-by-Step Solution:

1. **Use the identity for \( a^3 + b^3 \)**:
The formula for the sum of cubes is:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
We can also express \( a^2 - ab + b^2 \) in terms of \( a + b \) and \( ab \):
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Therefore,
\[ a^3 + b^3 = (a + b)((a + b)^2 - 3ab) \]

2. **Substituting the known value \( a + b = 3 \)**:
Substitute \( a + b = 3 \) into the equation:
\[ a^3 + b^3 = 3((3)^2 - 3ab) = 3(9 - 3ab) = 27 - 9ab \]

3. **Now, substitute \( a^3 + b^3 \) into the expression \( a^3 + b^3 + 9ab \)**:
\[ a^3 + b^3 + 9ab = (27 - 9ab) + 9ab = 27 \]

4. **Final Result**:
Thus, the value of \( a^3 + b^3 + 9ab \) is:
\[ \boxed{27} \]