If (a + b) = 6 and ab = 8 , then `(a^3+b^3)` is equal to

Options

A. 216
B. 144
C. 108
D. 72

Answer

To solve the problem, we need to find \(a^3 + b^3\) given that \(a + b = 6\) and \(ab = 8\).

### Step-by-step Solution:

1. **Use the identity for the sum of cubes**:
The formula for the sum of cubes is:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
We already know \(a + b = 6\) and \(ab = 8\).

2. **Find \(a^2 + b^2\)**:
We can find \(a^2 + b^2\) using the identity:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting the known values:
\[ a^2 + b^2 = (6)^2 - 2 \times 8 = 36 - 16 = 20 \]

3. **Substitute \(a^2 + b^2\) into the sum of cubes formula**:
Now we can substitute \(a^2 + b^2\) into the sum of cubes formula:
\[ a^3 + b^3 = (a + b)((a^2 + b^2) - ab) \]
Substituting the values we have:
\[ a^3 + b^3 = 6 \left(20 - 8\right) = 6 \times 12 = 72 \]

### Final Answer:
Thus, \(a^3 + b^3 = 72\).