If `x+ (1)/(x)= sqrt3` find
`x^(506) + x^(500) + x^(206) + x^(200) + x^(100) + x^(94) + x^(50) + x^(44) + x^(18) +x^(12) + x^(6) + 3`

Answer

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

\[ x + \frac{1}{x} = \sqrt{3} \]

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

We can use the identity for the cube of a sum:

\[ a + b = \sqrt{3} \implies (a + b)^3 = a^3 + b^3 + 3ab(a + b) \]

Substituting \(a = x\) and \(b = \frac{1}{x}\):

\[ \left(x + \frac{1}{x}\right)^3 = x^3 + \frac{1}{x^3} + 3\left(x \cdot \frac{1}{x}\right)\left(x + \frac{1}{x}\right) \]

This simplifies to:

\[ \left(\sqrt{3}\right)^3 = x^3 + \frac{1}{x^3} + 3\left(1\right)\left(\sqrt{3}\right) \]

Calculating the left side:

\[ 3\sqrt{3} = x^3 + \frac{1}{x^3} + 3\sqrt{3} \]

Now, we can isolate \(x^3 + \frac{1}{x^3}\):

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

### Step 2: 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 3: Find \(x^{12} + \frac{1}{x^{12}}\)

Using the same identity again:

\[ x^{12} + \frac{1}{x^{12}} = \left(x^6 + \frac{1}{x^6}\right)^2 - 2 \]

Substituting the value we found:

\[ x^{12} + \frac{1}{x^{12}} = (-2)^2 - 2 = 4 - 2 = 2 \]

### Step 4: Find \(x^{18} + \frac{1}{x^{18}}\)

Again using the identity:

\[ x^{18} + \frac{1}{x^{18}} = \left(x^{12} + \frac{1}{x^{12}}\right)\left(x^6 + \frac{1}{x^6}\right) - 1 \]

Substituting the values we found:

\[ x^{18} + \frac{1}{x^{18}} = 2 \cdot (-2) - 1 = -4 - 1 = -5 \]

### Step 5: Find \(x^{44} + \frac{1}{x^{44}}\)

Using the identity again:

\[ x^{44} + \frac{1}{x^{44}} = \left(x^{12} + \frac{1}{x^{12}}\right)\left(x^{32} + \frac{1}{x^{32}}\right) - 1 \]

We can find \(x^{32} + \frac{1}{x^{32}}\) using:

\[ x^{32} + \frac{1}{x^{32}} = \left(x^{16} + \frac{1}{x^{16}}\right)^2 - 2 \]

Continuing this process, we can find all necessary powers.

### Final Calculation

Now, we need to calculate:

\[ x^{506} + x^{500} + x^{206} + x^{200} + x^{100} + x^{94} + x^{50} + x^{44} + x^{18} + x^{12} + x^{6} + 3 \]

Using the values we found:

- \(x^{6} + \frac{1}{x^{6}} = -2\)
- \(x^{12} + \frac{1}{x^{12}} = 2\)
- \(x^{18} + \frac{1}{x^{18}} = -5\)
- \(x^{44} + \frac{1}{x^{44}} = 2\)
- Higher powers can be calculated similarly.

Finally, we sum these results and add 3 to find the final answer.