What is the value of `((i+sqrt(3))/(-i+sqrt(3)))^(200)+((i-sqrt(3))/(i+sqrt(3)))^(200) +1` ?

Options

A. A)`-1`
B. B)0
C. C)1
D. D)2

Answer

To solve the expression

\[ \left(\frac{i + \sqrt{3}}{-i + \sqrt{3}}\right)^{200} + \left(\frac{i - \sqrt{3}}{i + \sqrt{3}}\right)^{200} + 1, \]

we will follow these steps:

### Step 1: Simplify the first term

We start with the first term:

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

To simplify this, we multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{(i + \sqrt{3})(-i + \sqrt{3})}{(-i + \sqrt{3})(-i + \sqrt{3})}. \]

Calculating the denominator:

\[ (-i + \sqrt{3})(-i + \sqrt{3}) = (-i)^2 + 2(-i)(\sqrt{3}) + (\sqrt{3})^2 = 1 - 2i\sqrt{3} + 3 = 4 - 2i\sqrt{3}. \]

Now for the numerator:

\[ (i + \sqrt{3})(-i + \sqrt{3}) = -i^2 + i\sqrt{3} - i\sqrt{3} + 3 = 1 + 3 = 4. \]

Thus, we have:

\[ \frac{4}{4 - 2i\sqrt{3}}. \]

### Step 2: Rationalize the first term

Next, we rationalize:

\[ \frac{4}{4 - 2i\sqrt{3}} \cdot \frac{4 + 2i\sqrt{3}}{4 + 2i\sqrt{3}} = \frac{16 + 8i\sqrt{3}}{16 + 12} = \frac{16 + 8i\sqrt{3}}{28} = \frac{4 + 2i\sqrt{3}}{7}. \]

### Step 3: Simplify the second term

Now we simplify the second term:

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

Again, we multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{(i - \sqrt{3})(i - \sqrt{3})}{(i + \sqrt{3})(i - \sqrt{3})}. \]

Calculating the denominator:

\[ (i + \sqrt{3})(i - \sqrt{3}) = i^2 - (\sqrt{3})^2 = -1 - 3 = -4. \]

Now for the numerator:

\[ (i - \sqrt{3})(i - \sqrt{3}) = i^2 - 2i\sqrt{3} + 3 = -1 - 2i\sqrt{3} + 3 = 2 - 2i\sqrt{3}. \]

Thus, we have:

\[ \frac{2 - 2i\sqrt{3}}{-4} = -\frac{1 - i\sqrt{3}}{2}. \]

### Step 4: Combine the terms

Now we have:

\[ \left(\frac{4 + 2i\sqrt{3}}{7}\right)^{200} + \left(-\frac{1 - i\sqrt{3}}{2}\right)^{200} + 1. \]

Let \( \omega = \frac{4 + 2i\sqrt{3}}{7} \) and \( \omega^2 = -\frac{1 - i\sqrt{3}}{2} \).

### Step 5: Evaluate the powers

Since \( \omega \) and \( \omega^2 \) are roots of unity, we can express them in terms of \( \omega \) and \( \omega^2 \):

\[ \omega^{200} + \omega^{400} + 1. \]

Using the properties of roots of unity, we find:

\[ \omega^{200} = \omega^{0} = 1 \quad \text{and} \quad \omega^{400} = \omega^{0} = 1. \]

### Step 6: Final evaluation

Thus, we have:

\[ 1 + 1 + 1 = 3. \]

### Conclusion

The final value of the expression is:

\[ \boxed{3}. \]