If `x = sqrt5 + sqrt3 and y = sqrt5 - sqrt3,` then the value of `x ^(4) - y ^(4)` is

Options

A. 544
B. `32 sqrt15`
C. `64 sqrt15`
D. 384

Answer

To solve the problem, we need to find the value of \( x^4 - y^4 \) given that \( x = \sqrt{5} + \sqrt{3} \) and \( y = \sqrt{5} - \sqrt{3} \).

### Step-by-Step Solution:

1. **Identify the expressions for \( x \) and \( y \)**:
\[ x = \sqrt{5} + \sqrt{3} \]
\[ y = \sqrt{5} - \sqrt{3} \]

2. **Use the difference of squares formula**:
The expression \( x^4 - y^4 \) can be rewritten using the difference of squares:
\[ x^4 - y^4 = (x^2 + y^2)(x^2 - y^2) \]

3. **Calculate \( x^2 \) and \( y^2 \)**:
- For \( x^2 \):
\[ x^2 = (\sqrt{5} + \sqrt{3})^2 = 5 + 3 + 2\sqrt{15} = 8 + 2\sqrt{15} \]
- For \( y^2 \):
\[ y^2 = (\sqrt{5} - \sqrt{3})^2 = 5 + 3 - 2\sqrt{15} = 8 - 2\sqrt{15} \]

4. **Calculate \( x^2 + y^2 \)**:
\[ x^2 + y^2 = (8 + 2\sqrt{15}) + (8 - 2\sqrt{15}) = 8 + 8 = 16 \]

5. **Calculate \( x^2 - y^2 \)**:
\[ x^2 - y^2 = (8 + 2\sqrt{15}) - (8 - 2\sqrt{15}) = 2\sqrt{15} + 2\sqrt{15} = 4\sqrt{15} \]

6. **Combine the results**:
Now substitute back into the expression for \( x^4 - y^4 \):
\[ x^4 - y^4 = (x^2 + y^2)(x^2 - y^2) = 16 \cdot 4\sqrt{15} = 64\sqrt{15} \]

### Final Answer:
Thus, the value of \( x^4 - y^4 \) is:
\[ \boxed{64\sqrt{15}} \]