If `a-b = 3 and ab = 28,` then `a ^(3) - b ^(3)` is

Options

A. 279
B. 387
C. 512
D. 91

Answer

To solve the problem, we need to find the value of \( a^3 - b^3 \) given that \( a - b = 3 \) and \( ab = 28 \).

### Step 1: Use the identity for \( a^3 - b^3 \)
We can use the algebraic identity:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
From the problem, we know \( a - b = 3 \).

### 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:
- \( a - b = 3 \) so \( (a - b)^2 = 3^2 = 9 \)
- \( ab = 28 \)

Now substituting these into the equation:
\[ a^2 + b^2 = 9 + 2 \times 28 = 9 + 56 = 65 \]

### Step 3: Substitute into the identity for \( a^3 - b^3 \)
Now we can substitute \( a^2 + b^2 \) and \( ab \) into the identity:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
We already have:
- \( a - b = 3 \)
- \( a^2 + b^2 = 65 \)
- \( ab = 28 \)

So we can calculate:
\[ a^2 + ab + b^2 = a^2 + b^2 + ab = 65 + 28 = 93 \]

### Step 4: Calculate \( a^3 - b^3 \)
Now substituting back into the equation:
\[ a^3 - b^3 = 3 \times 93 = 279 \]

### Final Answer
Thus, the value of \( a^3 - b^3 \) is \( 279 \).

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