If `x + (( 1)/( x)) = sqrt( 13)` , then what is the value of `x^(5) - ((1)/( x^(5)))` ?

Options

A. 169
B. `169 sqrt(3)`
C. 393
D. 507

Answer

To solve the equation \( x + \frac{1}{x} = \sqrt{13} \) and find the value of \( x^5 - \frac{1}{x^5} \), we can follow these steps:

### Step 1: Square the initial equation
We start with the equation:
\[ x + \frac{1}{x} = \sqrt{13} \]
Now, we square both sides:
\[ \left( x + \frac{1}{x} \right)^2 = (\sqrt{13})^2 \]
This simplifies to:
\[ x^2 + 2 + \frac{1}{x^2} = 13 \]
Thus, we have:
\[ x^2 + \frac{1}{x^2} = 13 - 2 = 11 \]

### Step 2: Find \( x^3 - \frac{1}{x^3} \)
We can use the identity:
\[ x^3 - \frac{1}{x^3} = \left( x + \frac{1}{x} \right) \left( x^2 - 1 + \frac{1}{x^2} \right) \]
We already know \( x + \frac{1}{x} = \sqrt{13} \) and \( x^2 + \frac{1}{x^2} = 11 \). Therefore:
\[ x^2 - 1 + \frac{1}{x^2} = (x^2 + \frac{1}{x^2}) - 1 = 11 - 1 = 10 \]
Now substituting back:
\[ x^3 - \frac{1}{x^3} = \sqrt{13} \cdot 10 = 10\sqrt{13} \]

### Step 3: Find \( x^5 - \frac{1}{x^5} \)
We can use the identity:
\[ x^5 - \frac{1}{x^5} = \left( x^3 - \frac{1}{x^3} \right) \left( x^2 + 1 + \frac{1}{x^2} \right) \]
We already calculated \( x^3 - \frac{1}{x^3} = 10\sqrt{13} \) and we know \( x^2 + \frac{1}{x^2} = 11 \), so:
\[ x^2 + 1 + \frac{1}{x^2} = 11 + 1 = 12 \]
Now substituting back:
\[ x^5 - \frac{1}{x^5} = (10\sqrt{13}) \cdot 12 = 120\sqrt{13} \]

### Final Answer
Thus, the value of \( x^5 - \frac{1}{x^5} \) is:
\[ \boxed{120\sqrt{13}} \]