If ` x = sqrt(5) + sqrt(3) and y = sqrt( 5) - sqrt(3)` , then value of ` ( x^(4) - y^(4))` is

Options

A. `64 sqrt( 15)`
B. 16
C. 544
D. ` 32 sqrt( 15 )`

Answer

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

### Step 1: Identify the expression
We start with the expression \( x^4 - y^4 \).

### Step 2: Use the difference of squares
We can use the identity for the difference of squares:
\[ a^2 - b^2 = (a - b)(a + b) \]
In our case, we can rewrite \( x^4 - y^4 \) as:
\[ (x^2)^2 - (y^2)^2 = (x^2 - y^2)(x^2 + y^2) \]

### Step 3: Find \( x^2 \) and \( y^2 \)
Now, we need to calculate \( x^2 \) and \( y^2 \):
\[ x^2 = (\sqrt{5} + \sqrt{3})^2 = 5 + 3 + 2\sqrt{5}\sqrt{3} = 8 + 2\sqrt{15} \]
\[ y^2 = (\sqrt{5} - \sqrt{3})^2 = 5 + 3 - 2\sqrt{5}\sqrt{3} = 8 - 2\sqrt{15} \]

### Step 4: Calculate \( x^2 - y^2 \) and \( x^2 + y^2 \)
Now we can find \( x^2 - y^2 \) and \( x^2 + y^2 \):
\[ x^2 - y^2 = (8 + 2\sqrt{15}) - (8 - 2\sqrt{15}) = 4\sqrt{15} \]
\[ x^2 + y^2 = (8 + 2\sqrt{15}) + (8 - 2\sqrt{15}) = 16 \]

### Step 5: Substitute back into the expression
Now we can substitute back into our expression for \( x^4 - y^4 \):
\[ x^4 - y^4 = (x^2 - y^2)(x^2 + y^2) = (4\sqrt{15})(16) = 64\sqrt{15} \]

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