If `a-b=5 and ab=-4`, then find `a^(3)-b^(3)`.

Answer

To find \( a^3 - b^3 \) given that \( a - b = 5 \) and \( ab = -4 \), we can use the 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 know \( a - b = 5 \) and \( ab = -4 \). To use the identity, we first need to find \( a^2 + b^2 \).

Using the identity:

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

Substituting the known values:

\[ a^2 + b^2 = (5)^2 + 2(-4) \]

Calculating this gives:

\[ a^2 + b^2 = 25 - 8 = 17 \]

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

\[ a^2 + ab + b^2 = a^2 + b^2 + ab = 17 + (-4) = 13 \]

### Step 2: Substitute into the identity

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

\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) = 5 \cdot 13 \]

Calculating this gives:

\[ a^3 - b^3 = 65 \]

### Final Answer

Thus, the value of \( a^3 - b^3 \) is \( 65 \).

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