What is the value of `:`
`((-1+isqrt(3))/(2))^(3n) + (( -1-isqrt(3))/(2))^(3n)`
Where, `i= sqrt( -1) `?

Options

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

Answer

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

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

where \( i = \sqrt{-1} \).

### Step 1: Identify the complex numbers
We can rewrite the complex numbers in the expression:
- Let \( \omega = -1 + i\sqrt{3}/2 \)
- Let \( \omega^2 = -1 - i\sqrt{3}/2 \)

### Step 2: Recognize the cube roots of unity
We know that:
- \( \omega = e^{2\pi i / 3} \) (which corresponds to \( -1 + i\sqrt{3}/2 \))
- \( \omega^2 = e^{-2\pi i / 3} \) (which corresponds to \( -1 - i\sqrt{3}/2 \))

The cube roots of unity satisfy the equation:
\[ \omega^3 = 1 \quad \text{and} \quad 1 + \omega + \omega^2 = 0 \]

### Step 3: Substitute and simplify the expression
Using the cube roots of unity, we can rewrite the expression:
\[ \left(-1 + i\sqrt{3}/2\right)^{3n} + \left(-1 - i\sqrt{3}/2\right)^{3n} = \omega^{3n} + \omega^{2 \cdot 3n} \]

### Step 4: Evaluate the powers of \( \omega \)
Since \( \omega^3 = 1 \):
\[ \omega^{3n} = 1^n = 1 \quad \text{and} \quad \omega^{2 \cdot 3n} = (1^2)^n = 1 \]

### Step 5: Combine the results
Thus, we have:
\[ \omega^{3n} + \omega^{2 \cdot 3n} = 1 + 1 = 2 \]

### Final Answer
The value of the expression is:
\[ \boxed{2} \]

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