Hello Tolis.

Using

**Gauss Multiplication Formula** \[\displaystyle\prod_{k \mathop = 0}^{n - 1} \Gamma \Bigl({z + \frac k n}\Bigr) = ({2 \pi})^{\frac{n - 1}{2}} n^{\frac{1}{2} - n z} \Gamma({n z})\] for \(z=0\), we have that \begin{align*}

\prod_{k \mathop = 1}^{n - 1} \Gamma \Bigl({\frac k n}\Bigr)&=\prod_{k \mathop = 1}^{n - 1} \Gamma \Bigl({0 + \frac k n}\Bigr) \\

&= ({2 \pi})^{\frac{n - 1}{2}} n^{\frac{1}{2} - n \cdot0} \Gamma({n \cdot0})\\

&= ({2 \pi})^{\frac{n - 1}{2}} \,\sqrt{n}\,.

\end{align*}

A proof of the Gauss Multiplication Formula can be found in

here.