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

Options

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

Answer

To solve the problem step by step, we start with the equations provided:

1. **Given Equations**:
- \( x - y = 4 \) (Equation 1)
- \( xy = 45 \) (Equation 2)

2. **Using the Formula for \( x^3 - y^3 \)**:
The formula for the difference of cubes is:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
We already have \( x - y = 4 \). Now we need to find \( x^2 + xy + y^2 \).

3. **Finding \( x^2 + y^2 \)**:
We can express \( x^2 + y^2 \) using the identity:
\[ x^2 + y^2 = (x - y)^2 + 2xy \]
Substituting the values from our equations:
- \( (x - y)^2 = 4^2 = 16 \)
- \( 2xy = 2 \times 45 = 90 \)

Therefore:
\[ x^2 + y^2 = 16 + 90 = 106 \]

4. **Calculating \( x^2 + xy + y^2 \)**:
Now we can find \( x^2 + xy + y^2 \):
\[ x^2 + xy + y^2 = x^2 + y^2 + xy = 106 + 45 = 151 \]

5. **Substituting Back into the Formula**:
Now we substitute back into the formula for \( x^3 - y^3 \):
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) = 4 \times 151 \]

6. **Final Calculation**:
\[ x^3 - y^3 = 604 \]

Thus, the value of \( x^3 - y^3 \) is **604**.