If `(a-b)=4` and ab = 2, then `(a^(3)-b^(3))` is equal to :

Options

A. 92
B. 88
C. 84
D. 80

Answer

To solve the problem, we need to find the value of \( a^3 - b^3 \) given that \( a - b = 4 \) and \( ab = 2 \).

### Step-by-Step Solution:

1. **Recall the formula for the difference of cubes:**
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

2. **Substitute the known value of \( a - b \):**
We know that \( a - b = 4 \), so we can substitute this into the formula:
\[ a^3 - b^3 = 4(a^2 + ab + b^2) \]

3. **Find \( a^2 + ab + b^2 \):**
To find \( a^2 + ab + b^2 \), we can use the identity:
\[ a^2 + b^2 = (a - b)^2 + 2ab \]
First, calculate \( (a - b)^2 \):
\[ (a - b)^2 = 4^2 = 16 \]
Now substitute \( ab = 2 \):
\[ a^2 + b^2 = 16 + 2 \cdot 2 = 16 + 4 = 20 \]
Now, substitute \( ab = 2 \) into \( a^2 + ab + b^2 \):
\[ a^2 + ab + b^2 = a^2 + b^2 + ab = 20 + 2 = 22 \]

4. **Substitute back into the equation for \( a^3 - b^3 \):**
Now we can substitute \( a^2 + ab + b^2 = 22 \) back into the equation:
\[ a^3 - b^3 = 4 \cdot 22 = 88 \]

5. **Final answer:**
Thus, the value of \( a^3 - b^3 \) is \( 88 \).