Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2010 / B / 5 / d / iii

For $\kappa ={\aleph }_{\omega }$, monotonicity of cardinal exponentiation givesThe middle equality uses the currying law for cardinal exponentiation, and the final one uses ${\aleph }_0\cdot \kappa =\kappa$. Hence ${\aleph }_0^{\kappa }=2^{\kappa }$, even before assuming GCH. Applying the Generalized continuum hypothesis yields