= Solution
The product from the preceding part injects into $\kappa^{\aleph_0}$, where $\kappa=\aleph_\omega$, because each coordinate value is an <ordinal> below $\kappa$. Combining its strict König lower bound with the exponentiation upper bound gives
$$
\kappa<\kappa^{\aleph_0}\le\kappa^\kappa=\kappa^+.
$$
Hence
$$
\boxed{\aleph_\omega^{\aleph_0}=\aleph_{\omega+1}.}
$$
Only the instance $2^{\aleph_\omega}=\aleph_{\omega+1}$ of the <Generalized continuum hypothesis>[generalized continuum hypothesis] was needed in these three computations, though the question assumes the full hypothesis. No assumption that <countable> exponentiation leaves this <singular cardinal> unchanged is valid: König's theorem rules that out.
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