I know product of nth roots of unity is 1 or -1 depending whether n is odd or even.
But in this way I am getting 1. Where am I wrong?
$ \text{Let }\alpha = \cos \frac{2 \pi}{n} + \iota \sin \frac{2 \pi}{n} \text{ be a root of }x^n=1 \\
\text{Then product of nth roots will be } 1\cdot \alpha \cdot \alpha^2 … \alpha^{n-1} = \alpha^{\frac{(n)(n-1)}{2}}\\
=\left( \alpha^n \right)^{\frac{n-1}{2}}\\
=1^{\frac{n-1}{2}} \text{……By the definition of alpha ??}\\
=1 $
I can see this doesn't even work for n=2.
Best Answer
Consider the following: $$(-1)^1 = (-1)^{2\cdot \frac{1}{2}} = ((-1)^2)^\frac{1}{2}=1^{\frac{1}{2}} = 1$$ You did something like that. $n(n-1)/2$ is always an integer when $n$ is a natural number, but when you take out the $n$ you are left with $(n-1)/2$ which is not always an integer, which messes things up when $n$ is even. In short, the rule $a^{bc} = (a^b)^c$ does not always hold for complex numbers, in particular when $b$ or $c$ are not integers.