Ordinal numbers arithmetic $((1 + \omega) + \omega) \cdot \omega$ Cantor Normal Form

Question

how do you get the Cantor Normalform of $$((1 + \omega) + \omega) \cdot \omega$$ This would be $$((1 + \omega) + \omega) \cdot \omega = (\omega + \omega) \cdot \omega = (\omega \cdot 2) \cdot \omega$$ right? Is the attempt correct like this? How to proceed?

Answer 1

Here's a hint: Remember that ordinal multiplication is associative, so $(\omega\cdot2)\cdot\omega=\omega\cdot(2\cdot\omega)$, so you can simplify a bit more.