If a + b = 8, ab = -12, then `a^3 + b^3 =`

Options

A. `-244`
B. `-833`
C. `800`
D. `833`

Answer

To solve the problem, we need to find the value of \( a^3 + b^3 \) given that \( a + b = 8 \) and \( ab = -12 \).

### Step-by-Step Solution:

1. **Use the identity for \( a^3 + b^3 \)**:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

2. **Calculate \( a^2 + b^2 \)** using the identity:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substitute the known values:
\[ a^2 + b^2 = (8)^2 - 2(-12) \]
\[ a^2 + b^2 = 64 + 24 = 88 \]

3. **Substitute \( a^2 + b^2 \) back into the identity for \( a^3 + b^3 \)**:
\[ a^3 + b^3 = (a + b)((a^2 + b^2) - ab) \]
Substitute the known values:
\[ a^3 + b^3 = 8 \left( 88 - (-12) \right) \]
\[ a^3 + b^3 = 8 \left( 88 + 12 \right) = 8 \times 100 \]

4. **Calculate the final result**:
\[ a^3 + b^3 = 800 \]

Thus, the value of \( a^3 + b^3 \) is **800**.