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: Define Terms:
The imaginary cube roots of unity are \( \omega \) and \( \omega^2 \). They have the following properties: \( 1 + \omega + \omega^2 = 0 \) and \( \omega^3 = 1 \). Let \( a = \omega \).

Step 2: Key Properties:
From \( 1 + \omega + \omega^2 = 0 \), we derive: \( 1 + \omega^2 = -\omega \) and \( 1 + \omega = -\omega^2 \). These will be used to simplify terms within parentheses before exponentiation.

Step 3: Simplify and Calculate:
Substitute \( a = \omega \) into the expression.First term: \( (1 - \omega + \omega^2)^5 \)Rearrange: \( ((1+\omega^2) - \omega)^5 \)Substitute \( 1+\omega^2 = -\omega \): \( (-\omega - \omega)^5 = (-2\omega)^5 = -32\omega^5 \).Since \( \omega^5 = \omega^3 \cdot \omega^2 = 1 \cdot \omega^2 = \omega^2 \), the first term is \( -32\omega^2 \).Second term: \( (1 + \omega - \omega^2)^5 \)Rearrange: \( ((1+\omega) - \omega^2)^5 \)Substitute \( 1+\omega = -\omega^2 \): \( (-\omega^2 - \omega^2)^5 = (-2\omega^2)^5 = -32\omega^{10} \).Since \( \omega^{10} = (\omega^3)^3 \cdot \omega = 1^3 \cdot \omega = \omega \), the second term is \( -32\omega \).Sum the terms: \( -32\omega^2 + (-32\omega) = -32(\omega^2 + \omega) \).Using \( \omega + \omega^2 = -1 \) from \( 1 + \omega + \omega^2 = 0 \), the expression simplifies to \( -32(-1) = 32 \).
Step 4: Final Result:
The expression evaluates to 32.