The value of `a^(3)-b^(3)`, when `a-b=4andab=-2` is

Options

A. 88
B. 40
C. 72
D. 64

Answer

To find the value of \( a^3 - b^3 \) given that \( a - b = 4 \) and \( ab = -2 \), we can use the algebraic identity for the difference of cubes:

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

### Step 1: Calculate \( a^2 + ab + b^2 \)

We can express \( a^2 + ab + b^2 \) in terms of \( a - b \) and \( ab \).

We know that:

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

Substituting \( a - b = 4 \) and \( ab = -2 \):

\[ a^2 + b^2 = (4)^2 + 2(-2) \]

Calculating this gives:

\[ a^2 + b^2 = 16 - 4 = 12 \]

Now, we can find \( a^2 + ab + b^2 \):

\[ a^2 + ab + b^2 = a^2 + b^2 + ab = 12 + (-2) = 10 \]

### Step 2: Substitute values into the identity

Now we can substitute \( a - b \) and \( a^2 + ab + b^2 \) into the identity:

\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) = 4 \times 10 \]

Calculating this gives:

\[ a^3 - b^3 = 40 \]

### Final Answer

Thus, the value of \( a^3 - b^3 \) is \( 40 \).

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