The value of `((-1+isqrt( 3))/(2))^(n) + (( -1-isqrt(3))/(2))^(n)` where n is not a multiple of 3 and `i= sqrt(-1)` is `:`

Options

A. A)1
B. B)`-1`
C. C)i
D. D)`-i`

Answer

To solve the expression \(\left(\frac{-1 + i\sqrt{3}}{2}\right)^n + \left(\frac{-1 - i\sqrt{3}}{2}\right)^n\) where \(n\) is not a multiple of 3 and \(i = \sqrt{-1}\), we can follow these steps:

### Step 1: Identify the complex numbers
Let:
\[ z_1 = \frac{-1 + i\sqrt{3}}{2} \]
\[ z_2 = \frac{-1 - i\sqrt{3}}{2} \]

### Step 2: Convert to polar form
To express \(z_1\) and \(z_2\) in polar form, we need to find their modulus and argument.

**Modulus:**
\[ |z_1| = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]
\[ |z_2| = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{1} = 1 \]

**Argument:**
For \(z_1\):
\[ \theta_1 = \tan^{-1}\left(\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\right) = \tan^{-1}(-\sqrt{3}) = \frac{2\pi}{3} \quad (\text{in the second quadrant}) \]

For \(z_2\):
\[ \theta_2 = \tan^{-1}\left(\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \quad (\text{in the third quadrant}) \]

### Step 3: Write in polar form
Thus, we can express:
\[ z_1 = e^{i\frac{2\pi}{3}}, \quad z_2 = e^{-i\frac{2\pi}{3}} \]

### Step 4: Raise to the power \(n\)
Now we compute:
\[ z_1^n = \left(e^{i\frac{2\pi}{3}}\right)^n = e^{i\frac{2\pi n}{3}} \]
\[ z_2^n = \left(e^{-i\frac{2\pi}{3}}\right)^n = e^{-i\frac{2\pi n}{3}} \]

### Step 5: Combine the results
Now we can add these two results:
\[ z_1^n + z_2^n = e^{i\frac{2\pi n}{3}} + e^{-i\frac{2\pi n}{3}} = 2\cos\left(\frac{2\pi n}{3}\right) \]

### Step 6: Determine the value based on \(n\)
Since \(n\) is not a multiple of 3, \(\frac{2\pi n}{3}\) will not be an integer multiple of \(2\pi\), which means \(\cos\left(\frac{2\pi n}{3}\right)\) will not equal 1 or -1.

Thus, the expression simplifies to:
\[ 2\cos\left(\frac{2\pi n}{3}\right) \]

### Final Answer
The value of the expression is:
\[ 2\cos\left(\frac{2\pi n}{3}\right) \]