If x = `(sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3))`, find the value of `x^(3) + (1)/(x^(3))` .

Options

A. 644
B. 512
C. 488
D. 348

Answer

To solve the problem, we need to find the value of \( x^3 + \frac{1}{x^3} \) given that \( x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \).

### Step 1: Rationalize the expression for \( x \)

We start with:
\[ x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \]

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ x = \frac{(\sqrt{5} + \sqrt{3})(\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})} \]

### Step 2: Simplify the numerator and denominator

The numerator simplifies as follows:
\[ (\sqrt{5} + \sqrt{3})^2 = 5 + 3 + 2\sqrt{15} = 8 + 2\sqrt{15} \]

The denominator simplifies using the difference of squares:
\[ (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \]

Thus, we have:
\[ x = \frac{8 + 2\sqrt{15}}{2} = 4 + \sqrt{15} \]

### Step 3: Find \( \frac{1}{x} \)

Now we find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{4 + \sqrt{15}} \]

To rationalize this, we multiply the numerator and denominator by the conjugate:
\[ \frac{1}{x} = \frac{4 - \sqrt{15}}{(4 + \sqrt{15})(4 - \sqrt{15})} \]

Calculating the denominator:
\[ (4)^2 - (\sqrt{15})^2 = 16 - 15 = 1 \]

Thus:
\[ \frac{1}{x} = 4 - \sqrt{15} \]

### Step 4: Calculate \( x + \frac{1}{x} \)

Now we can find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (4 + \sqrt{15}) + (4 - \sqrt{15}) = 8 \]

### Step 5: Use the identity to find \( x^3 + \frac{1}{x^3} \)

We use the identity:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3\left( x + \frac{1}{x} \right) \]

Substituting \( x + \frac{1}{x} = 8 \):
\[ x^3 + \frac{1}{x^3} = 8^3 - 3 \cdot 8 \]

Calculating \( 8^3 \):
\[ 8^3 = 512 \]

Calculating \( 3 \cdot 8 \):
\[ 3 \cdot 8 = 24 \]

Thus:
\[ x^3 + \frac{1}{x^3} = 512 - 24 = 488 \]

### Final Answer

The value of \( x^3 + \frac{1}{x^3} \) is:
\[ \boxed{488} \]