If ` ( x + (1)/( x)) = 3 sqrt(2)` then what is the value of ` ( x^(5) + (1)/( x^(5)))` ?

Options

A. `178 sqrt(3)`
B. `789 sqrt(2)`
C. `1581 sqrt(2)`
D. `717 sqrt(2)`

Answer

To solve the problem, we need to find the value of \( x^5 + \frac{1}{x^5} \) given that \( x + \frac{1}{x} = 3\sqrt{2} \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x + \frac{1}{x} = 3\sqrt{2} \]

2. **Find \( x^2 + \frac{1}{x^2} \):**
We can use the identity:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
Substituting the value we have:
\[ x^2 + \frac{1}{x^2} = (3\sqrt{2})^2 - 2 = 18 - 2 = 16 \]

3. **Find \( x^3 + \frac{1}{x^3} \):**
We can use the identity:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right) \left( x^2 + \frac{1}{x^2} \right) - \left( x + \frac{1}{x} \right) \]
Substituting the values we have:
\[ x^3 + \frac{1}{x^3} = (3\sqrt{2})(16) - (3\sqrt{2}) = 48\sqrt{2} - 3\sqrt{2} = 45\sqrt{2} \]

4. **Find \( x^5 + \frac{1}{x^5} \):**
We can use the identity:
\[ x^5 + \frac{1}{x^5} = \left( x^3 + \frac{1}{x^3} \right) \left( x^2 + \frac{1}{x^2} \right) - \left( x + \frac{1}{x} \right) \]
Substituting the values we have:
\[ x^5 + \frac{1}{x^5} = (45\sqrt{2})(16) - (3\sqrt{2}) = 720\sqrt{2} - 3\sqrt{2} = 717\sqrt{2} \]

### Final Answer:
\[ x^5 + \frac{1}{x^5} = 717\sqrt{2} \]