Question

The value of \((\frac{-1+i\sqrt{3}}{2})^{18}+(\frac{-1-i\sqrt{3}}{2})^{18}\) is

• A. \(2\)
• B. \(35\)
• C. \(\sqrt{3}\)
• D. \(\frac{\sqrt{3}}{2}\)

Hint:

The two numbers are the complex cube roots of unity, so use w cubed equals 1.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The numbers \(\omega = \frac{-1+i\sqrt{3}}{2}\) and \(\omega^2 = \frac{-1-i\sqrt{3}}{2}\) are the two non-real cube roots of unity. They satisfy \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\).

Step 2: Key Formula or Approach:
Write each power of \(\omega\) as a power of \(\omega^3\) and use \(\omega^3 = 1\).

Step 3: Detailed Explanation:
The first term is \(\omega^{18} = (\omega^3)^6 = 1^6 = 1\).
The second term is \((\omega^2)^{18} = \omega^{36} = (\omega^3)^{12} = 1\).
\[ \omega^{18} + \omega^{36} = 1 + 1 = 2 \]
The values \(35\), \(\sqrt{3}\) and \(\frac{\sqrt{3}}{2}\) do not arise, because every power of a cube root of unity is again \(1\), \(\omega\) or \(\omega^2\).

Final Answer:
The value is \(2\), option (A). \[ \boxed{2} \]