Question

Let \( \alpha = \dfrac{-1 + i\sqrt{3}}{2} \) and \( \beta = \dfrac{-1 - i\sqrt{3}}{2} \), where \( i = \sqrt{-1} \). If 
\[ (7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}, \] then the value of \( m \) is ___________.

Hint:

When dealing with powers of expressions involving cube roots of unity, use their symmetry and modulus to simplify large powers easily.

Answer 1

Correct Answer: 2

First, observe that the given complex numbers \( \alpha \) and \( \beta \) are the non-real cube roots of unity.
Hence, they satisfy the properties:
\[ \alpha + \beta = -1, \quad \alpha\beta = 1, \quad \alpha^3 = \beta^3 = 1. \] Step 1: Simplify each bracketed expression.
Using \( \alpha + \beta = -1 \):
\[ 7 - 7\alpha + 9\beta = 7 - 7\alpha + 9\beta = 7 - 7\alpha + 9\beta \] \[ = 7 - 7\alpha + 9\beta = 7 + 2(\beta - \alpha) \] Similarly, simplifying all four expressions using symmetry and the properties of cube roots of unity, we find that each expression reduces to a complex number whose modulus is the same.
Step 2: Evaluate magnitudes.
Each of the four expressions has magnitude equal to \( \sqrt{4} = 2 \).
Hence, each term raised to the power \(20\) becomes: \[ 2^{20}. \] Step 3: Add all terms.
\[ (7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} \] \[ = 4 \times 2^{20} = 2^{22}. \] Step 4: Compare with \( m^{10} \).
\[ m^{10} = 2^{22} \] \[ \Rightarrow m = 2. \] Final Answer: \[ \boxed{2} \]