for x=3 and y=-2 verify that
`(x-y)^(3)=x^(3)-y^(3)-3xy(x-y)`

Answer

To verify the equation \((x - y)^3 = x^3 - y^3 - 3xy(x - y)\) for \(x = 3\) and \(y = -2\), we will calculate both the left-hand side (LHS) and the right-hand side (RHS) of the equation step by step.

### Step 1: Calculate the Left-Hand Side (LHS)
The left-hand side of the equation is:
\[ LHS = (x - y)^3 \]
Substituting the values of \(x\) and \(y\):
\[ LHS = (3 - (-2))^3 \]
This simplifies to:
\[ LHS = (3 + 2)^3 = 5^3 \]
Calculating \(5^3\):
\[ LHS = 125 \]

### Step 2: Calculate the Right-Hand Side (RHS)
The right-hand side of the equation is:
\[ RHS = x^3 - y^3 - 3xy(x - y) \]
Substituting the values of \(x\) and \(y\):
\[ RHS = 3^3 - (-2)^3 - 3 \cdot 3 \cdot (-2) \cdot (3 - (-2)) \]
Calculating \(3^3\) and \((-2)^3\):
\[ 3^3 = 27 \quad \text{and} \quad (-2)^3 = -8 \]
Now substituting these values into the RHS:
\[ RHS = 27 - (-8) - 3 \cdot 3 \cdot (-2) \cdot (3 + 2) \]
This simplifies to:
\[ RHS = 27 + 8 - 3 \cdot 3 \cdot (-2) \cdot 5 \]
Calculating \(27 + 8\):
\[ RHS = 35 - 3 \cdot 3 \cdot (-2) \cdot 5 \]
Calculating \(3 \cdot 3 \cdot (-2) \cdot 5\):
\[ 3 \cdot 3 = 9 \quad \text{and} \quad 9 \cdot (-2) = -18 \]
Thus:
\[ -18 \cdot 5 = -90 \]
Now substituting this back into the RHS:
\[ RHS = 35 - (-90) = 35 + 90 = 125 \]

### Step 3: Compare LHS and RHS
Now we have:
\[ LHS = 125 \quad \text{and} \quad RHS = 125 \]
Since \(LHS = RHS\), we can conclude that the equation is verified.

### Final Conclusion
Thus, we have verified that:
\[ (x - y)^3 = x^3 - y^3 - 3xy(x - y) \quad \text{for} \quad x = 3 \quad \text{and} \quad y = -2 \]