Find the value of pq if `p^(3)-q^(3)=68andp-q=-4`.

Answer

To find the value of \( pq \) given the equations \( p^3 - q^3 = 68 \) and \( p - q = -4 \), we can use the algebraic identity for the difference of cubes:

\[ p^3 - q^3 = (p - q)(p^2 + pq + q^2) \]

### Step 1: Substitute the value of \( p - q \)
We know that \( p - q = -4 \). We can substitute this into the identity:

\[ p^3 - q^3 = (-4)(p^2 + pq + q^2) \]

### Step 2: Set up the equation
Now we can set this equal to the other equation:

\[ 68 = (-4)(p^2 + pq + q^2) \]

### Step 3: Divide both sides by -4
To isolate \( p^2 + pq + q^2 \), divide both sides by -4:

\[ p^2 + pq + q^2 = \frac{68}{-4} = -17 \]

### Step 4: Use the identity for \( (p - q)^2 \)
We can also express \( p^2 + q^2 \) in terms of \( p - q \):

\[ (p - q)^2 = p^2 - 2pq + q^2 \]

Since \( p - q = -4 \):

\[ (-4)^2 = p^2 - 2pq + q^2 \]

This simplifies to:

\[ 16 = p^2 - 2pq + q^2 \]

### Step 5: Substitute \( p^2 + q^2 \)
Now we can express \( p^2 + q^2 \) in terms of \( pq \):

\[ p^2 + q^2 = 16 + 2pq \]

### Step 6: Substitute back into the equation
Now we can substitute \( p^2 + q^2 \) into our earlier equation:

\[ 16 + 2pq + pq = -17 \]

This simplifies to:

\[ 16 + 3pq = -17 \]

### Step 7: Isolate \( pq \)
Now, isolate \( pq \):

\[ 3pq = -17 - 16 \]
\[ 3pq = -33 \]

### Step 8: Solve for \( pq \)
Finally, divide both sides by 3:

\[ pq = \frac{-33}{3} = -11 \]

### Final Answer
Thus, the value of \( pq \) is:

\[ \boxed{-11} \]