= Solution
For <GCH arithmetic at aleph omega>, let $\kappa=\aleph_\omega$. Use monotonicity of <cardinal exponentiation>, the law $(a^b)^c=a^{bc}$, and $\kappa\cdot\kappa=\kappa$ for infinite cardinals in <ZFC>. Since $2\le\kappa\le2^\kappa$,
$$
2^\kappa\le\kappa^\kappa\le(2^\kappa)^\kappa=2^{\kappa\cdot\kappa}=2^\kappa.
$$
Thus $\kappa^\kappa=2^\kappa$, and the <Generalized continuum hypothesis>[generalized continuum hypothesis] yields
$$
\boxed{\aleph_\omega^{\aleph_\omega}=\aleph_{\omega+1}.}
$$
The lower bound embeds binary-valued <functions> into $\kappa$-valued ones; the upper bound codes each value by a <subset> and combines the resulting pairs of indices. In particular the equality before applying the hypothesis is a <ZFC> equality.
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