If x- y = 7 and ` x^(3) - y^(3) = 133. ` find :
` xy `

Answer

To solve the problem where \( x - y = 7 \) and \( x^3 - y^3 = 133 \), we can use the identity for the difference of cubes:

\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]

### Step 1: Substitute the known values into the identity
We know that \( x - y = 7 \) and \( x^3 - y^3 = 133 \). Substituting these values into the identity gives us:

\[ 133 = 7(x^2 + xy + y^2) \]

### Step 2: Simplify the equation
Now, we can divide both sides of the equation by 7:

\[ x^2 + xy + y^2 = \frac{133}{7} \]

Calculating \( \frac{133}{7} \):

\[ x^2 + xy + y^2 = 19 \]

### Step 3: Use the square of the difference
We can also use the square of the difference \( (x - y)^2 \) to express \( x^2 + y^2 \):

\[ (x - y)^2 = x^2 - 2xy + y^2 \]

Substituting \( x - y = 7 \):

\[ 7^2 = x^2 - 2xy + y^2 \]

Calculating \( 7^2 \):

\[ 49 = x^2 - 2xy + y^2 \]

### Step 4: Express \( x^2 + y^2 \) in terms of \( xy \)
We can express \( x^2 + y^2 \) as follows:

\[ x^2 + y^2 = (x^2 - 2xy + y^2) + 2xy = 49 + 2xy \]

### Step 5: Substitute back into the equation
Now we can substitute \( x^2 + y^2 \) into our earlier equation:

\[ 49 + 2xy + xy = 19 \]

This simplifies to:

\[ 49 + 3xy = 19 \]

### Step 6: Solve for \( xy \)
Now, we can isolate \( xy \):

\[ 3xy = 19 - 49 \]
\[ 3xy = -30 \]
\[ xy = \frac{-30}{3} \]
\[ xy = -10 \]

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

\[ \boxed{-10} \]