If `x+ (1)/(x) = sqrt3` find
`x^(117) + (1)/(x^(117))`

Answer

To solve the problem, we start with the given equation:

### Step 1: Start with the equation
Given:
\[ x + \frac{1}{x} = \sqrt{3} \]

### Step 2: Cube both sides
We will cube both sides of the equation:
\[ \left(x + \frac{1}{x}\right)^3 = \left(\sqrt{3}\right)^3 \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} + 3\left(x + \frac{1}{x}\right) = 3\sqrt{3} \]

### Step 3: Substitute the value of \(x + \frac{1}{x}\)
Substituting \(\sqrt{3}\) into the equation:
\[ x^3 + \frac{1}{x^3} + 3\sqrt{3} = 3\sqrt{3} \]

### Step 4: Solve for \(x^3 + \frac{1}{x^3}\)
Now, we can isolate \(x^3 + \frac{1}{x^3}\):
\[ x^3 + \frac{1}{x^3} = 3\sqrt{3} - 3\sqrt{3} = 0 \]

### Step 5: Find \(x^{117} + \frac{1}{x^{117}}\)
Since we know \(x^3 + \frac{1}{x^3} = 0\), we can express \(x^{117} + \frac{1}{x^{117}}\) in terms of powers of \(x^3\):
\[ x^{117} + \frac{1}{x^{117}} = \left(x^3\right)^{39} + \frac{1}{\left(x^3\right)^{39}} = x^{117} + \frac{1}{x^{117}} \]

### Step 6: Use the property of powers
Using the property of powers:
\[ x^{117} + \frac{1}{x^{117}} = \left(x^3\right)^{39} + \frac{1}{\left(x^3\right)^{39}} = 0 \]

### Final Answer
Thus, the final result is:
\[ x^{117} + \frac{1}{x^{117}} = 0 \]

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