I am trying to calculate $\kappa^\lambda = \aleph_{\omega_1}^{\aleph_0}$.
I know that if $\kappa$ is a limit cardinal and $0 < \lambda < \mathrm{cf}(\kappa)$ then $\kappa^{\lambda} = \displaystyle \sum_{\alpha < \kappa} |\alpha|^{\lambda}$.
Hence $ \aleph_{\omega_1}^{\aleph_0} = \displaystyle \sum_{\alpha < \aleph_{\omega_1}} |\alpha|^{\aleph_0}$.
I also know that if $\kappa, \lambda$ are infinite cardinals then $(\kappa^+)^\lambda = \kappa^\lambda \cdot \kappa^+$ so that $\aleph_\alpha^{\aleph_0} = \aleph_\alpha \cdot \aleph_{\alpha-1}^{\aleph_0} = \aleph_\alpha \cdot \aleph_{\alpha-1} \cdot \aleph_{\alpha-2}^{\aleph_0} = \dots = \aleph_\alpha \cdot \aleph_{\alpha-1} \cdot \dots \cdot \aleph_{0}^{\aleph_0} = \aleph_\alpha$.
Hence $ \aleph_{\omega_1}^{\aleph_0} = \displaystyle \sum_{\alpha < \aleph_{\omega_1}} |\alpha|^{\aleph_0} = \sum_{\alpha < \aleph_{\omega_1}} |\alpha| = \sum_{\alpha < \omega_1} \aleph_{\alpha}$.
Since for infinite cardinals $\lambda \le \kappa$ we have that $\lambda + \kappa = \kappa$, $\displaystyle \sum_{\alpha < \omega_1} \aleph_{\alpha} = \sup_{\alpha < \omega_1} \aleph_{\alpha} = \aleph_{\omega_1}$.
Hence $\aleph_{\omega_1}^{\aleph_0} = \aleph_{\omega_1}$. Is this correct? This is an exercise in Just/Weese and the hint is "Assume GCH". I don't think I have used GCH so I suspect I am missing something. Thanks for your help.
Edit
I have used a different method to compute it and have reached the same result (although I still don't know whether this is correct):
By Tarski's theorem, $\aleph_{\omega_1}^{\aleph_0} = \displaystyle \sum_{\alpha < \aleph_{\omega_1}} |\alpha|^{\aleph_0}$.
Since $|\alpha|^{\aleph_0} \le \aleph_{\omega_1}$ for all $\alpha < \aleph_{\omega_1}$ we get $\displaystyle \sum_{\alpha < \aleph_{\omega_1}} |\alpha|^{\aleph_0} \leq \aleph_{\omega_1}$. Of course, $\aleph_{\omega_1} \leq \aleph_{\omega_1}^{\aleph_0}$.
Hence $\aleph_{\omega_1} = \aleph_{\omega_1}^{\aleph_0}$.