If `x + (1)/(x) = sqrt(5)`, then `x^(3) + (1)/(x^(3))` is equal to :

Options

A. A)`3sqrt(5)`
B. B)`4sqrt(5)`
C. C)`2sqrt(5)`
D. D)`5sqrt(5)`

Answer

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

### Step-by-Step Solution:

1. **Given Equation**: Start with the equation provided in the problem:
\[ x + \frac{1}{x} = \sqrt{5} \]

2. **Cube Both Sides**: We will use the identity for cubes:
\[ (a + b)^3 = a^3 + b^3 + 3ab(a + b) \]
Here, let \( a = x \) and \( b = \frac{1}{x} \). Thus,
\[ (x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3(x)(\frac{1}{x})(x + \frac{1}{x}) \]
This simplifies to:
\[ (x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x}) \]

3. **Substitute the Known Value**: Substitute \( x + \frac{1}{x} = \sqrt{5} \):
\[ (\sqrt{5})^3 = x^3 + \frac{1}{x^3} + 3\sqrt{5} \]

4. **Calculate \( (\sqrt{5})^3 \)**: Calculate \( \sqrt{5} \) cubed:
\[ (\sqrt{5})^3 = 5\sqrt{5} \]

5. **Set Up the Equation**: Now we have:
\[ 5\sqrt{5} = x^3 + \frac{1}{x^3} + 3\sqrt{5} \]

6. **Isolate \( x^3 + \frac{1}{x^3} \)**: Rearranging gives:
\[ x^3 + \frac{1}{x^3} = 5\sqrt{5} - 3\sqrt{5} \]

7. **Simplify**: This simplifies to:
\[ x^3 + \frac{1}{x^3} = (5 - 3)\sqrt{5} = 2\sqrt{5} \]

### Final Answer:
Thus, the value of \( x^3 + \frac{1}{x^3} \) is:
\[ \boxed{2\sqrt{5}} \]