Question

If \((9+7\alpha-7\beta)^{20} + (9\alpha+7\beta-7)^{20} + (9\beta+7-7\alpha)^{20} + (14+7\alpha+7\beta)^{20}\) is \(m^{10}\) then the value of m is : (where \(\alpha = \frac{-1+i\sqrt{3}}{2}\) \& \(\beta = \frac{-1-i\sqrt{3}}{2}\))

• A. 50
• B. 49
• C. 46
• D. 48

Hint:

In problems involving sums of powers of expressions with cube roots of unity, look for symmetric relations. Check if the terms are related by multiplication of \(\omega\) or \(\omega^2\). This often leads to a massive simplification using the property \(1+\omega+\omega^2=0\).

Answer 1

The Correct Option is B

Concept: We are given the following expression: \[ (9 + 7\alpha - 7\beta)^{20} + (9\alpha + 7\beta - 7)^{20} + (9\beta + 7 - 7\alpha)^{20} + (14 + 7\alpha + 7\beta)^{20} \] where \(\alpha = \frac{-1 + i\sqrt{3}}{2}\) and \(\beta = \frac{-1 - i\sqrt{3}}{2}\). We need to find the value of \(m\) such that this expression equals \(m^{10}\).  
Step 1: Simplifying the expression using the properties of \(\alpha\) and \(\beta\). Note that \(\alpha\) and \(\beta\) are complex cube roots of unity, satisfying the following properties: \[ \alpha^2 + \alpha + 1 = 0 \quad \text{and} \quad \beta^2 + \beta + 1 = 0. \] Additionally: \[ \alpha + \beta = -1 \quad \text{and} \quad \alpha \beta = 1. \] 
Step 2: Analyze each term in the given expression. Let’s substitute \(\alpha + \beta = -1\) and \(\alpha \beta = 1\) into the terms of the expression. - For the term \((9 + 7\alpha - 7\beta)^{20}\): \[ 9 + 7\alpha - 7\beta = 9 + 7(\alpha - \beta). \] Since \(\alpha - \beta = \left( \frac{-1 + i\sqrt{3}}{2} \right) - \left( \frac{-1 - i\sqrt{3}}{2} \right) = i\sqrt{3}\), we have: \[ 9 + 7\alpha - 7\beta = 9 + 7i\sqrt{3}. \] Thus, the term becomes: \[ (9 + 7i\sqrt{3})^{20}. \] - Similarly, for the other terms: - \((9\alpha + 7\beta - 7)^{20}\) becomes: \[ (9\alpha + 7\beta - 7) = 9\alpha + 7\beta - 7 = 7(\alpha + \beta) = 7(-1) = -7. \] So, this term simplifies to: \[ (-7)^{20}. \] - \((9\beta + 7 - 7\alpha)^{20}\) becomes: \[ (9\beta + 7 - 7\alpha) = 9\beta + 7 - 7\alpha = 7(\beta + \alpha) = 7(-1) = -7. \] Thus, this term simplifies to: \[ (-7)^{20}. \] - \((14 + 7\alpha + 7\beta)^{20}\) becomes: \[ (14 + 7\alpha + 7\beta) = 14 + 7(\alpha + \beta) = 14 + 7(-1) = 7. \] So, this term simplifies to: \[ 7^{20}. \] 
Therefore, \(m = 49\)