If `x= sqrt5+sqrt3` and `y = sqrt5-sqrt3`, then `x^4 -y^4`

Explanation

To solve the problem where \( x = \sqrt{5} + \sqrt{3} \) and \( y = \sqrt{5} - \sqrt{3} \), we need to find \( x^4 - y^4 \). We can use the difference of squares and some algebraic identities to simplify the calculations.

### Step-by-Step Solution:

1. **Express \( x^4 - y^4 \) using the difference of squares:**
\[ x^4 - y^4 = (x^2 - y^2)(x^2 + y^2) \]

**Hint:** Recall that \( a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) \).

2. **Calculate \( x^2 - y^2 \):**
\[ x^2 - y^2 = (x - y)(x + y) \]

**Hint:** Use the identity \( a^2 - b^2 = (a - b)(a + b) \).

3. **Find \( x - y \) and \( x + y \):**
- \( x - y = (\sqrt{5} + \sqrt{3}) - (\sqrt{5} - \sqrt{3}) = 2\sqrt{3} \)
- \( x + y = (\sqrt{5} + \sqrt{3}) + (\sqrt{5} - \sqrt{3}) = 2\sqrt{5} \)

**Hint:** Simplify the expressions by combining like terms.

4. **Now substitute back into \( x^2 - y^2 \):**
\[ x^2 - y^2 = (2\sqrt{3})(2\sqrt{5}) = 4\sqrt{15} \]

**Hint:** Multiply the results from step 3.

5. **Next, calculate \( x^2 + y^2 \):**
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
First, find \( xy \):
\[ xy = (\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3}) = 5 - 3 = 2 \]

**Hint:** Use the difference of squares to find \( xy \).

6. **Now calculate \( (x + y)^2 \):**
\[ (x + y)^2 = (2\sqrt{5})^2 = 4 \cdot 5 = 20 \]

**Hint:** Square the sum you found in step 3.

7. **Substituting into \( x^2 + y^2 \):**
\[ x^2 + y^2 = 20 - 2 \cdot 2 = 20 - 4 = 16 \]

**Hint:** Perform the subtraction carefully.

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

**Hint:** Multiply the results from steps 4 and 7.

### Final Answer:
\[ x^4 - y^4 = 64\sqrt{15} \]