What is the value of `((1=-+isqrt3)/(2))^(3n)+((-1+isqrt3)/(2))^(3n) where i=sqrt(-1)`?

Options

A. 3
B. 2
C. 1
D. 0

Explanation

To solve the problem, we need to evaluate the expression:

\[ \left(\frac{-1 - i\sqrt{3}}{2}\right)^{3n} + \left(\frac{-1 + i\sqrt{3}}{2}\right)^{3n} \]

### Step 1: Identify the cube roots of unity

The terms \(\frac{-1 - i\sqrt{3}}{2}\) and \(\frac{-1 + i\sqrt{3}}{2}\) can be recognized as the cube roots of unity. We denote:

\[ \omega = \frac{-1 + i\sqrt{3}}{2} \quad \text{and} \quad \omega^2 = \frac{-1 - i\sqrt{3}}{2} \]

### Step 2: Rewrite the expression using \(\omega\)

Now we can rewrite the original expression using \(\omega\):

\[ \left(\omega^2\right)^{3n} + \left(\omega\right)^{3n} \]

### Step 3: Simplify the powers

Using the property of exponents, we simplify:

\[ \left(\omega^2\right)^{3n} = \omega^{6n} \quad \text{and} \quad \left(\omega\right)^{3n} = \omega^{3n} \]

Thus, the expression becomes:

\[ \omega^{6n} + \omega^{3n} \]

### Step 4: Use the property of cube roots of unity

We know that \(\omega^3 = 1\). Therefore:

\[ \omega^{6n} = (\omega^3)^{2n} = 1^{2n} = 1 \]
\[ \omega^{3n} = (\omega^3)^{n} = 1^{n} = 1 \]

### Step 5: Combine the results

Now, we can combine these results:

\[ 1 + 1 = 2 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{2} \]

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