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

Options

A. 488
B. 504
C. 472
D. 476

Answer

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

### Step 1: Calculate \( y \)
Since \( y \) is the reciprocal of \( x \), we have:
\[ y = \frac{1}{x} = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \]

### Step 2: Use the identity for \( x^3 + y^3 \)
We can use the identity:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
To use this identity, we need to find \( x + y \) and \( xy \).

### Step 3: Calculate \( x + y \)
\[ x + y = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} + \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \]
To add these fractions, we need a common denominator:
\[ x + y = \frac{(\sqrt{5} - \sqrt{3})^2 + (\sqrt{5} + \sqrt{3})^2}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})} \]

### Step 4: Simplify the numerator
Calculating the squares:
\[ (\sqrt{5} - \sqrt{3})^2 = 5 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15} \]
\[ (\sqrt{5} + \sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15} \]
Adding these:
\[ (\sqrt{5} - \sqrt{3})^2 + (\sqrt{5} + \sqrt{3})^2 = (8 - 2\sqrt{15}) + (8 + 2\sqrt{15}) = 16 \]

### Step 5: Calculate the denominator
The denominator simplifies as follows:
\[ (\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3}) = 5 - 3 = 2 \]

### Step 6: Combine results
Thus,
\[ x + y = \frac{16}{2} = 8 \]

### Step 7: Calculate \( xy \)
\[ xy = x \cdot y = \left(\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}\right) \cdot \left(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\right) = 1 \]

### Step 8: Calculate \( x^2 + y^2 \)
Using the identity \( x^2 + y^2 = (x + y)^2 - 2xy \):
\[ x^2 + y^2 = 8^2 - 2 \cdot 1 = 64 - 2 = 62 \]

### Step 9: Substitute into the identity for \( x^3 + y^3 \)
Now substituting into the identity:
\[ x^3 + y^3 = (x + y)((x^2 + y^2) - xy) = 8(62 - 1) = 8 \cdot 61 = 488 \]

### Final Answer
Thus, the value of \( x^3 + y^3 \) is:
\[ \boxed{488} \]