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: Find \( x^2 + \frac{1}{x^2} \)

We start with the given equation:
\[ x + \frac{1}{x} = \sqrt{3} \]

We square both sides:
\[ \left(x + \frac{1}{x}\right)^2 = (\sqrt{3})^2 \]
This expands to:
\[ x^2 + 2 + \frac{1}{x^2} = 3 \]
Subtracting 2 from both sides gives:
\[ x^2 + \frac{1}{x^2} = 1 \]

### Step 2: Find \( x^3 + \frac{1}{x^3} \)

Using 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 known values:
\[ x^3 + \frac{1}{x^3} = \left(\sqrt{3}\right)(1) - \sqrt{3} = \sqrt{3} - \sqrt{3} = 0 \]

### Step 3: Find \( x^6 + \frac{1}{x^6} \)

Using 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 \( x^{12} + \frac{1}{x^{12}} \)

Using the identity:
\[ x^{12} + \frac{1}{x^{12}} = \left(x^6 + \frac{1}{x^6}\right)^2 - 2 \]
Substituting the value we found:
\[ x^{12} + \frac{1}{x^{12}} = (-2)^2 - 2 = 4 - 2 = 2 \]

### Step 5: Find \( x^{17} + \frac{1}{x^{17}} \)

Using the identity:
\[ x^{17} + \frac{1}{x^{17}} = \left(x^{12} + \frac{1}{x^{12}}\right)\left(x^5 + \frac{1}{x^5}\right) - \left(x^{7} + \frac{1}{x^{7}}\right) \]

First, we need \( x^5 + \frac{1}{x^5} \) and \( x^7 + \frac{1}{x^7} \).

### Step 6: Find \( x^4 + \frac{1}{x^4} \)

Using the identity:
\[ x^4 + \frac{1}{x^4} = \left(x^2 + \frac{1}{x^2}\right)^2 - 2 \]
Substituting the value we found:
\[ x^4 + \frac{1}{x^4} = 1^2 - 2 = 1 - 2 = -1 \]

### Step 7: Find \( x^5 + \frac{1}{x^5} \)

Using the identity:
\[ x^5 + \frac{1}{x^5} = \left(x^4 + \frac{1}{x^4}\right)\left(x + \frac{1}{x}\right) - \left(x^3 + \frac{1}{x^3}\right) \]
Substituting the values:
\[ x^5 + \frac{1}{x^5} = (-1)(\sqrt{3}) - 0 = -\sqrt{3} \]

### Step 8: Find \( x^7 + \frac{1}{x^7} \)

Using the identity:
\[ x^7 + \frac{1}{x^7} = \left(x^6 + \frac{1}{x^6}\right)\left(x + \frac{1}{x}\right) - \left(x^5 + \frac{1}{x^5}\right) \]
Substituting the values:
\[ x^7 + \frac{1}{x^7} = (-2)(\sqrt{3}) - (-\sqrt{3}) = -2\sqrt{3} + \sqrt{3} = -\sqrt{3} \]

### Step 9: Calculate \( x^{17} + \frac{1}{x^{17}} \)

Now we can substitute back into our equation:
\[ x^{17} + \frac{1}{x^{17}} = \left(2\right)(-\sqrt{3}) - (-\sqrt{3}) = -2\sqrt{3} + \sqrt{3} = -\sqrt{3} \]

Thus, the final answer is:
\[ \boxed{-\sqrt{3}} \]