If `(x-y)=4` and `xy = 45`, then the value of `x^(3) - y^(3)` is:

Options

A. 82
B. 604
C. 822
D. 151

Answer

To solve the problem, we need to find the value of \( x^3 - y^3 \) given that \( x - y = 4 \) and \( xy = 45 \).

### Step-by-step Solution:

1. **Use the identity for the difference of cubes**:
The formula for the difference of cubes is:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
We already know \( x - y = 4 \).

2. **Calculate \( x^2 + xy + y^2 \)**:
We need to find \( x^2 + xy + y^2 \). We can express \( x^2 + y^2 \) in terms of \( (x - y)^2 \) and \( xy \):
\[ x^2 + y^2 = (x - y)^2 + 2xy \]
Substituting the known values:
\[ x^2 + y^2 = (4)^2 + 2 \cdot 45 = 16 + 90 = 106 \]
Now, we can find \( x^2 + xy + y^2 \):
\[ x^2 + xy + y^2 = x^2 + y^2 + xy = 106 + 45 = 151 \]

3. **Substitute back into the difference of cubes formula**:
Now we can substitute back into the formula for \( x^3 - y^3 \):
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) = 4 \cdot 151 = 604 \]

4. **Final Answer**:
Therefore, the value of \( x^3 - y^3 \) is \( 604 \).

### Conclusion:
The correct option is \( 604 \).