If `x+ (1)/(x) = sqrt3` find
`x^(25) + (1)/(x^(25))`

Answer

To solve the problem where \( x + \frac{1}{x} = \sqrt{3} \) and we need to find \( x^{25} + \frac{1}{x^{25}} \), we can follow these steps:

### Step 1: Find \( x^2 + \frac{1}{x^2} \)
We start with the identity:
\[ \left( x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2} \]
Substituting \( x + \frac{1}{x} = \sqrt{3} \):
\[ \left( \sqrt{3} \right)^2 = x^2 + 2 + \frac{1}{x^2} \]
\[ 3 = x^2 + 2 + \frac{1}{x^2} \]
Rearranging gives:
\[ x^2 + \frac{1}{x^2} = 3 - 2 = 1 \]

### 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:
\[ x^3 + \frac{1}{x^3} = \sqrt{3} \cdot 1 - \sqrt{3} = 0 \]

### 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 \( x^3 + \frac{1}{x^3} = 0 \):
\[ x^6 + \frac{1}{x^6} = 0^2 - 2 = -2 \]

### Step 4: Find \( x^9 + \frac{1}{x^9} \)
Using the identity:
\[ x^9 + \frac{1}{x^9} = \left( x^6 + \frac{1}{x^6} \right) \left( x^3 + \frac{1}{x^3} \right) - \left( x^3 + \frac{1}{x^3} \right) \]
Substituting the known values:
\[ x^9 + \frac{1}{x^9} = (-2)(0) - 0 = 0 \]

### Step 5: 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 \( x^6 + \frac{1}{x^6} = -2 \):
\[ x^{12} + \frac{1}{x^{12}} = (-2)^2 - 2 = 4 - 2 = 2 \]

### Step 6: Find \( x^{15} + \frac{1}{x^{15}} \)
Using the identity:
\[ x^{15} + \frac{1}{x^{15}} = \left( x^{12} + \frac{1}{x^{12}} \right) \left( x^3 + \frac{1}{x^3} \right) - \left( x^3 + \frac{1}{x^3} \right) \]
Substituting the known values:
\[ x^{15} + \frac{1}{x^{15}} = (2)(0) - 0 = 0 \]

### Step 7: Find \( x^{24} + \frac{1}{x^{24}} \)
Using the identity:
\[ x^{24} + \frac{1}{x^{24}} = \left( x^{12} + \frac{1}{x^{12}} \right)^2 - 2 \]
Substituting \( x^{12} + \frac{1}{x^{12}} = 2 \):
\[ x^{24} + \frac{1}{x^{24}} = 2^2 - 2 = 4 - 2 = 2 \]

### Step 8: Find \( x^{25} + \frac{1}{x^{25}} \)
Using the identity:
\[ x^{25} + \frac{1}{x^{25}} = \left( x^{24} + \frac{1}{x^{24}} \right) \left( x + \frac{1}{x} \right) - \left( x^{23} + \frac{1}{x^{23}} \right) \]
We know \( x^{24} + \frac{1}{x^{24}} = 2 \) and \( x + \frac{1}{x} = \sqrt{3} \), and from previous calculations \( x^{23} + \frac{1}{x^{23}} = 0 \):
\[ x^{25} + \frac{1}{x^{25}} = 2 \cdot \sqrt{3} - 0 = 2\sqrt{3} \]

Thus, the final answer is:
\[ \boxed{2\sqrt{3}} \]