Given sum of a and b is 3. Then, find the value of `a^3 + 9ab + b^3`.

Options

A. 9
B. 24
C. 18
D. none of the above

Answer

To find the value of \( a^3 + 9ab + b^3 \) given that \( a + b = 3 \), we can follow these steps:

### Step 1: Use the identity for \( a^3 + b^3 \)
We know that:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Given \( a + b = 3 \), we can substitute this into the equation.

### Step 2: Find \( a^2 + b^2 \)
We can express \( a^2 + b^2 \) in terms of \( a + b \) and \( ab \):
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting \( a + b = 3 \):
\[ a^2 + b^2 = 3^2 - 2ab = 9 - 2ab \]

### Step 3: Substitute into the identity
Now we can substitute \( a^2 + b^2 \) back into the identity for \( a^3 + b^3 \):
\[ a^3 + b^3 = 3(9 - 2ab) - ab \]
This simplifies to:
\[ a^3 + b^3 = 27 - 6ab - ab = 27 - 7ab \]

### Step 4: Combine with \( 9ab \)
Now we need to find \( a^3 + 9ab + b^3 \):
\[ a^3 + 9ab + b^3 = (a^3 + b^3) + 9ab \]
Substituting the expression we found for \( a^3 + b^3 \):
\[ a^3 + 9ab + b^3 = (27 - 7ab) + 9ab \]
This simplifies to:
\[ a^3 + 9ab + b^3 = 27 + 2ab \]

### Step 5: Determine the value of \( ab \)
Since we don't have a specific value for \( ab \), we can express the final answer in terms of \( ab \):
\[ a^3 + 9ab + b^3 = 27 + 2ab \]

### Conclusion
Without additional information about \( ab \), we cannot find a numeric answer. However, we can conclude that:
\[ a^3 + 9ab + b^3 = 27 + 2ab \]