If ` a - b = 2 and ab = 24` then what is the value of ` (a^(3) - b^(3))` ?

Options

A. 280
B. 124
C. 140
D. 152

Answer

To solve the problem where \( a - b = 2 \) and \( ab = 24 \), and we need to find the value of \( a^3 - b^3 \), we can use the formula for the difference of cubes:

\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

### Step 1: Identify \( a - b \)
From the problem, we know:
\[ a - b = 2 \]

### Step 2: Find \( a^2 + b^2 \)
To find \( a^2 + b^2 \), we can use the identity:
\[ a^2 + b^2 = (a - b)^2 + 2ab \]
Substituting the known values:
\[ a^2 + b^2 = (2)^2 + 2(24) \]
Calculating this gives:
\[ a^2 + b^2 = 4 + 48 = 52 \]

### Step 3: Calculate \( a^2 + ab + b^2 \)
Now we can find \( a^2 + ab + b^2 \):
\[ a^2 + ab + b^2 = a^2 + b^2 + ab \]
Substituting the values we have:
\[ a^2 + ab + b^2 = 52 + 24 = 76 \]

### Step 4: Substitute into the difference of cubes formula
Now we can substitute back into the formula for \( a^3 - b^3 \):
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) = 2 \times 76 \]
Calculating this gives:
\[ a^3 - b^3 = 152 \]

### Final Answer
Thus, the value of \( a^3 - b^3 \) is \( 152 \).