`((1+sqrt(3)i)/(1-sqrt(3)i))^(181) + ((1-sqrt(3)i)/(1+sqrt(3)i))^(181)` is equal to ______________

Answer

To solve the expression

\[ \left(\frac{1+\sqrt{3}i}{1-\sqrt{3}i}\right)^{181} + \left(\frac{1-\sqrt{3}i}{1+\sqrt{3}i}\right)^{181}, \]

we can follow these steps:

### Step 1: Rewrite the Complex Numbers
We can express the complex numbers in terms of \( \omega \) and \( \omega^2 \), where \( \omega = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \) and \( \omega^2 = -\frac{1}{2} - \frac{\sqrt{3}}{2} i \), which are the cube roots of unity.

### Step 2: Simplify the Fractions
Divide the numerator and denominator of each fraction by 2:

\[ \frac{1+\sqrt{3}i}{1-\sqrt{3}i} = \frac{\frac{1+\sqrt{3}i}{2}}{\frac{1-\sqrt{3}i}{2}}. \]

Let \( z_1 = \frac{1+\sqrt{3}i}{2} \) and \( z_2 = \frac{1-\sqrt{3}i}{2} \).

### Step 3: Identify the Powers
Notice that:

\[ z_1 = \omega \quad \text{and} \quad z_2 = \omega^2. \]

Thus, we can rewrite the expression as:

\[ \left(\frac{z_1}{z_2}\right)^{181} + \left(\frac{z_2}{z_1}\right)^{181}. \]

### Step 4: Simplify Further
This can be expressed as:

\[ \left(\frac{\omega}{\omega^2}\right)^{181} + \left(\frac{\omega^2}{\omega}\right)^{181} = \omega^{181} + \omega^{-181}. \]

### Step 5: Calculate the Powers
Since \( \omega^3 = 1 \), we can reduce the exponent 181 modulo 3:

\[ 181 \mod 3 = 1. \]

Thus,

\[ \omega^{181} = \omega^1 = \omega \quad \text{and} \quad \omega^{-181} = \omega^{-1} = \omega^2. \]

### Step 6: Combine the Results
Now we have:

\[ \omega + \omega^2. \]

### Step 7: Use the Property of Roots of Unity
From the property of cube roots of unity:

\[ 1 + \omega + \omega^2 = 0 \implies \omega + \omega^2 = -1. \]

### Conclusion
Thus, the value of the original expression is:

\[ \boxed{-1}. \]