If `a^(2) + b^(2) = 99` and `ab = 11, (a gt 0, b gt 0)` then the value of `(a^(3) + b^(3))` is :

Options

A. 1250
B. 968
C. 1100
D. 1080

Answer

To solve the problem, we need to find the value of \( a^3 + b^3 \) given that \( a^2 + b^2 = 99 \) and \( ab = 11 \).

### Step-by-Step Solution:

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

2. **Find \( a + b \)**:
We know:
\[ a^2 + b^2 = 99 \]
We can use the identity:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting the known values:
\[ 99 = (a + b)^2 - 2 \cdot 11 \]
Simplifying this gives:
\[ 99 = (a + b)^2 - 22 \]
\[ (a + b)^2 = 99 + 22 = 121 \]
Taking the square root:
\[ a + b = \sqrt{121} = 11 \]

3. **Calculate \( a^2 - ab + b^2 \)**:
We already have \( a^2 + b^2 = 99 \) and \( ab = 11 \):
\[ a^2 - ab + b^2 = (a^2 + b^2) - ab = 99 - 11 = 88 \]

4. **Substitute into the identity**:
Now we can substitute \( a + b \) and \( a^2 - ab + b^2 \) into the formula for \( a^3 + b^3 \):
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) = 11 \cdot 88 \]
\[ a^3 + b^3 = 968 \]

### Final Answer:
The value of \( a^3 + b^3 \) is \( 968 \).