Question

If 'a' is an imaginary cube root of unity, then \( (1-a+a^2)^5 + (1+a-a^2)^5 \) is equal to:

• A. 4
• B. 5
• C. 32
• D. 16

Hint:

When dealing with expressions involving cube roots of unity (\(\omega, \omega^2\)), always look to use the identity \(1+\omega+\omega^2=0\) to simplify terms inside brackets before expanding. This almost always simplifies the problem dramatically.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The imaginary cube roots of unity are \( \omega \) and \( \omega^2 \). They satisfy two fundamental properties: 1. \( 1 + \omega + \omega^2 = 0 \) 2. \( \omega^3 = 1 \) We can let the imaginary cube root 'a' be \( \omega \).

Step 2: Key Formula or Approach:
We will use the properties of the cube roots of unity to simplify the expressions inside the parentheses before raising them to the power of 5. From \( 1 + \omega + \omega^2 = 0 \), we can derive: - \( 1 + \omega^2 = -\omega \) - \( 1 + \omega = -\omega^2 \)

Step 3: Detailed Explanation:
Let's substitute \( a = \omega \) into the given expression. First term: \( (1 - \omega + \omega^2)^5 \) Group the terms: \( ((1+\omega^2) - \omega)^5 \) Substitute \( 1+\omega^2 = -\omega \): \[ (-\omega - \omega)^5 = (-2\omega)^5 = (-2)^5 (\omega)^5 = -32 \omega^5 \] Since \( \omega^3=1 \), we can simplify \( \omega^5 = \omega^3 \cdot \omega^2 = 1 \cdot \omega^2 = \omega^2 \). So the first term is \( -32\omega^2 \). Second term: \( (1 + \omega - \omega^2)^5 \) Group the terms: \( ((1+\omega) - \omega^2)^5 \) Substitute \( 1+\omega = -\omega^2 \): \[ (-\omega^2 - \omega^2)^5 = (-2\omega^2)^5 = (-2)^5 (\omega^2)^5 = -32 \omega^{10} \] Simplify \( \omega^{10} = (\omega^3)^3 \cdot \omega = (1)^3 \cdot \omega = \omega \). So the second term is \( -32\omega \). Now, add the two simplified terms: \[ \text{Expression} = -32\omega^2 + (-32\omega) = -32(\omega^2 + \omega) \] From the property \( 1 + \omega + \omega^2 = 0 \), we know that \( \omega + \omega^2 = -1 \). \[ \text{Expression} = -32(-1) = 32 \]
Step 4: Final Answer:
The value of the expression is 32.