If `x=3sqrt(5)+2sqrt(2)` and `y=3sqrt(5)-2sqrt(2)` , then the value of `(x^(2)-y^(2))^(2)` is

Options

A. `240`
B. `140`
C. `5760`
D. `5300`

Answer

To solve the problem step by step, we start with the given values of \( x \) and \( y \):

1. **Given Values:**
\[ x = 3\sqrt{5} + 2\sqrt{2} \]
\[ y = 3\sqrt{5} - 2\sqrt{2} \]

2. **Using the Identity:**
We need to find \( (x^2 - y^2)^2 \). We can use the identity \( a^2 - b^2 = (a + b)(a - b) \). Thus, we can rewrite our expression as:
\[ x^2 - y^2 = (x + y)(x - y) \]

3. **Calculating \( x + y \):**
\[ x + y = (3\sqrt{5} + 2\sqrt{2}) + (3\sqrt{5} - 2\sqrt{2}) \]
Simplifying this:
\[ x + y = 3\sqrt{5} + 3\sqrt{5} + 2\sqrt{2} - 2\sqrt{2} = 6\sqrt{5} \]

4. **Calculating \( x - y \):**
\[ x - y = (3\sqrt{5} + 2\sqrt{2}) - (3\sqrt{5} - 2\sqrt{2}) \]
Simplifying this:
\[ x - y = 3\sqrt{5} - 3\sqrt{5} + 2\sqrt{2} + 2\sqrt{2} = 4\sqrt{2} \]

5. **Substituting Back:**
Now substituting \( x + y \) and \( x - y \) back into the expression:
\[ x^2 - y^2 = (6\sqrt{5})(4\sqrt{2}) \]

6. **Calculating the Product:**
\[ x^2 - y^2 = 24\sqrt{10} \]

7. **Finding \( (x^2 - y^2)^2 \):**
Now we need to square the result:
\[ (x^2 - y^2)^2 = (24\sqrt{10})^2 \]
\[ = 576 \times 10 = 5760 \]

Thus, the final answer is:
\[ \boxed{5760} \]