`(sqrt(x)+sqrt(y))^(2)=x+y+2sqrt(xy)` and `sqrt(x)sqrt(y)=sqrt(xy)` , where `x` and `y` are positive real numbers .
If `x=2sqrt(5)+sqrt(3)` and `y=2sqrt(5)-sqrt(3)`, then `x^(4)+y^(4)` =

Options

A. `1538`
B. `1200`
C. `1048`
D. `149`

Answer

To solve the problem, we need to calculate \( x^4 + y^4 \) given \( x = 2\sqrt{5} + \sqrt{3} \) and \( y = 2\sqrt{5} - \sqrt{3} \).

### Step 1: Calculate \( x^2 \) and \( y^2 \)

We start by calculating \( x^2 \):
\[ x^2 = (2\sqrt{5} + \sqrt{3})^2 = (2\sqrt{5})^2 + 2(2\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 \]
Calculating each term:
- \( (2\sqrt{5})^2 = 4 \cdot 5 = 20 \)
- \( 2(2\sqrt{5})(\sqrt{3}) = 4\sqrt{15} \)
- \( (\sqrt{3})^2 = 3 \)

Thus,
\[ x^2 = 20 + 4\sqrt{15} + 3 = 23 + 4\sqrt{15} \]

Now, we calculate \( y^2 \):
\[ y^2 = (2\sqrt{5} - \sqrt{3})^2 = (2\sqrt{5})^2 - 2(2\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 \]
Calculating each term:
- \( (2\sqrt{5})^2 = 20 \)
- \( -2(2\sqrt{5})(\sqrt{3}) = -4\sqrt{15} \)
- \( (\sqrt{3})^2 = 3 \)

Thus,
\[ y^2 = 20 - 4\sqrt{15} + 3 = 23 - 4\sqrt{15} \]

### Step 2: Calculate \( x^2 + y^2 \)

Now, we add \( x^2 \) and \( y^2 \):
\[ x^2 + y^2 = (23 + 4\sqrt{15}) + (23 - 4\sqrt{15}) = 46 \]

### Step 3: Calculate \( x^2y^2 \)

Next, we calculate \( x^2y^2 \):
\[ x^2y^2 = (x^2)(y^2) = (23 + 4\sqrt{15})(23 - 4\sqrt{15}) \]
This is a difference of squares:
\[ x^2y^2 = 23^2 - (4\sqrt{15})^2 = 529 - 240 = 289 \]

### Step 4: Calculate \( x^4 + y^4 \)

Now we can use the identity:
\[ x^4 + y^4 = (x^2 + y^2)^2 - 2x^2y^2 \]
Substituting the values we found:
\[ x^4 + y^4 = 46^2 - 2 \cdot 289 = 2116 - 578 = 1538 \]

Thus, the final answer is:
\[ \boxed{1538} \]