Question

If \(\alpha_1,\alpha_2,\ldots,\alpha_{23}\) are the \(23^{rd}\) roots of unity, then \[ \alpha_1^{47}+\alpha_2^{47}+\cdots+\alpha_{23}^{47}= \]

• A. \(23\)
• B. \(-1\)
• C. \(1\)
• D. \(0\)

Hint:

For \(n^{th}\) roots of unity, the sum of \(m^{th}\) powers of all roots is \(0\) when \(n\nmid m\), and is \(n\) when \(n\mid m\).

Answer 1

The Correct Option is D

Step 1: Use the property of roots of unity.
The \(23^{rd}\) roots of unity are the roots of \[ x^{23}=1 \] These roots are \[ \alpha_1,\alpha_2,\ldots,\alpha_{23} \] For \(23^{rd}\) roots of unity, we know that \[ \alpha_1^m+\alpha_2^m+\cdots+\alpha_{23}^m=0 \] if \(23\nmid m\).

Step 2: Check the exponent.
Here, \[ m=47 \] Now, \[ 47=23\times 2+1 \] So, \[ 23\nmid 47 \] Therefore, \[ \alpha_1^{47}+\alpha_2^{47}+\cdots+\alpha_{23}^{47}=0 \]

Step 3: Final conclusion.
Hence, \[ \boxed{0} \]