<s>[INST] How do you multiply this How can you multiply these ordinal numbers:
$(\omega+1)(\omega+1)(\omega2+2)$
I tried and have gotten to this:
$(\omega^2+1)(\omega2+2)$
Is that the correct way, or did i made a mistake?
 [/INST] Ordinal multiplication is distributive in the rightmost element. Therefore 
$$(\omega+1)(\omega+1)=(\omega+1)\omega+(\omega+1)=\omega^2+\omega+1$$
Therefore:
\begin{align*}
(\omega+1)(\omega+1)(\omega\cdot2+2) &=(\omega+1)(\omega+1)\omega\cdot2+(\omega+1)(\omega+1)2 \\ &=(\omega+1)(\omega+1)\omega+(\omega+1)(\omega+1)\omega+(\omega+1)(\omega+1)2 \\ &= (\omega^2+\omega+1)\omega+(\omega^2+\omega+1)\omega+(\omega^2+\omega+1)+(\omega^2+\omega+1) \\ &= \omega^3+\omega^3+\omega^2+\omega+1+\omega^2+\omega+1 \\ &=\omega^3 \cdot 2+ \omega^2 \cdot 2 + \omega + 1 \end{align*}
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