If ` a + b = 3 ` then the value of ` a^(3) + b^(3) + 9ab - 27` is

Options

A. 24
B. 25
C. 0
D. 27

Answer

To solve the problem, we need to find the value of \( a^3 + b^3 + 9ab - 27 \) given that \( a + b = 3 \).

### Step-by-Step Solution:

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

2. **Substitute \( a + b \)**:
Since \( a + b = 3 \), we can substitute this into our equations:
\[ a^3 + b^3 = 3(a^2 - ab + b^2) \]

3. **Express \( a^2 + b^2 \)**:
Using the expression for \( a^2 + b^2 \):
\[ a^2 + b^2 = (3)^2 - 2ab = 9 - 2ab \]
Therefore:
\[ a^3 + b^3 = 3((9 - 2ab) - ab) = 3(9 - 3ab) = 27 - 9ab \]

4. **Combine with the original expression**:
Now we substitute \( a^3 + b^3 \) into the expression we want to evaluate:
\[ a^3 + b^3 + 9ab - 27 = (27 - 9ab) + 9ab - 27 \]

5. **Simplify the expression**:
Simplifying the above expression:
\[ = 27 - 9ab + 9ab - 27 = 0 \]

Thus, the value of \( a^3 + b^3 + 9ab - 27 \) is \( \boxed{0} \).