If `x+ (1)/(x)= -sqrt3` find
`x^(17) + (1)/(x^(17))`

Answer

To solve the problem where \( x + \frac{1}{x} = -\sqrt{3} \) and we need to find \( x^{17} + \frac{1}{x^{17}} \), we can follow these steps:

### Step 1: Cube the given equation
We start with the equation:
\[ x + \frac{1}{x} = -\sqrt{3} \]
Now, we will cube both sides:
\[ \left( x + \frac{1}{x} \right)^3 = (-\sqrt{3})^3 \]
Calculating the right side:
\[ (-\sqrt{3})^3 = -3\sqrt{3} \]

### Step 2: Apply the cube expansion formula
Using the formula \( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \):
\[ x^3 + \frac{1}{x^3} + 3 \left( x \cdot \frac{1}{x} \right) \left( x + \frac{1}{x} \right) = -3\sqrt{3} \]
Since \( x \cdot \frac{1}{x} = 1 \), we can simplify:
\[ x^3 + \frac{1}{x^3} + 3(-\sqrt{3}) = -3\sqrt{3} \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} - 3\sqrt{3} = -3\sqrt{3} \]
Thus:
\[ x^3 + \frac{1}{x^3} = -3\sqrt{3} + 3\sqrt{3} = 0 \]

### Step 3: Find \( x^6 + \frac{1}{x^6} \)
Now we use the identity:
\[ x^6 + \frac{1}{x^6} = \left( x^3 + \frac{1}{x^3} \right)^2 - 2 \]
Substituting the value we found:
\[ x^6 + \frac{1}{x^6} = 0^2 - 2 = -2 \]

### Step 4: Find \( x^{12} + \frac{1}{x^{12}} \)
Next, we find \( x^{12} + \frac{1}{x^{12}} \) using:
\[ x^{12} + \frac{1}{x^{12}} = \left( x^6 + \frac{1}{x^6} \right)^2 - 2 \]
Substituting the value we found:
\[ x^{12} + \frac{1}{x^{12}} = (-2)^2 - 2 = 4 - 2 = 2 \]

### Step 5: Find \( x^{17} + \frac{1}{x^{17}} \)
Now, we can find \( x^{17} + \frac{1}{x^{17}} \) using:
\[ x^{17} + \frac{1}{x^{17}} = \left( x^{12} + \frac{1}{x^{12}} \right) \left( x^5 + \frac{1}{x^5} \right) - \left( x^{7} + \frac{1}{x^{7}} \right) \]
First, we need \( x^5 + \frac{1}{x^5} \) and \( x^7 + \frac{1}{x^7} \).

### Step 6: Find \( x^5 + \frac{1}{x^5} \)
Using the identity:
\[ x^5 + \frac{1}{x^5} = (x^3 + \frac{1}{x^3})(x^2 + \frac{1}{x^2}) - (x + \frac{1}{x}) \]
We need \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2 = (-\sqrt{3})^2 - 2 = 3 - 2 = 1 \]
Now substituting:
\[ x^5 + \frac{1}{x^5} = (0)(1) - (-\sqrt{3}) = \sqrt{3} \]

### Step 7: Find \( x^7 + \frac{1}{x^7} \)
Using:
\[ x^7 + \frac{1}{x^7} = (x^6 + \frac{1}{x^6})(x + \frac{1}{x}) - (x^5 + \frac{1}{x^5}) \]
Substituting:
\[ x^7 + \frac{1}{x^7} = (-2)(-\sqrt{3}) - \sqrt{3} = 2\sqrt{3} - \sqrt{3} = \sqrt{3} \]

### Step 8: Final calculation for \( x^{17} + \frac{1}{x^{17}} \)
Now substituting back:
\[ x^{17} + \frac{1}{x^{17}} = (2)(\sqrt{3}) - (\sqrt{3}) = 2\sqrt{3} - \sqrt{3} = \sqrt{3} \]

### Final Answer
Thus, the value of \( x^{17} + \frac{1}{x^{17}} \) is:
\[ \boxed{\sqrt{3}} \]