If `a^(2) + b^(2) = 135` and `ab = 7, (a gt 0, b gt 0)` then the value of `(a^(3) - b^(3))` is :

Options

A. 1562
B. 1600
C. 1680
D. 1350

Answer

To solve the problem, we need to find the value of \( a^3 - b^3 \) given the equations \( a^2 + b^2 = 135 \) and \( ab = 7 \).

### 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 \( a^2 + b^2 \)**:
We already know from the problem statement that:
\[ a^2 + b^2 = 135 \]

3. **Calculate \( a^2 + ab + b^2 \)**:
We can express \( a^2 + ab + b^2 \) as:
\[ a^2 + ab + b^2 = a^2 + b^2 + ab = 135 + 7 = 142 \]

4. **Find \( a - b \)**:
We can use the identity:
\[ a^2 + b^2 = (a - b)^2 + 2ab \]
Plugging in the known values:
\[ 135 = (a - b)^2 + 2 \cdot 7 \]
Simplifying gives:
\[ 135 = (a - b)^2 + 14 \]
\[ (a - b)^2 = 135 - 14 = 121 \]
Taking the square root:
\[ a - b = \sqrt{121} = 11 \]

5. **Substituting back to find \( a^3 - b^3 \)**:
Now we have \( a - b = 11 \) and \( a^2 + ab + b^2 = 142 \). Thus:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) = 11 \cdot 142 \]
Calculating this gives:
\[ a^3 - b^3 = 1562 \]

### Final Answer:
The value of \( a^3 - b^3 \) is \( 1562 \).