Leading term of an ordinal

The leading term ${\omega }^{\delta }n$ is the highest-exponent term in Cantor normal form; the remaining tail is strictly below ${\omega }^{\delta }$.

Every nonzero ordinal $\alpha$ has a greatest exponent $\beta$ such that ${\omega }^{\beta }\le \alpha$. It is the leading exponent in the Cantor normal form of $\alpha$, andfor a positive finite integer $n$ and $\gamma <{\omega }^{\beta }$.

Let nonzero ordinals $\xi ,\eta$ have leading exponents $\rho ,\sigma$. If $\eta$ is finite, then $\xi \eta$ has leading exponent $\rho$. If $\eta$ is infinite, so that $\sigma >0$, then $\xi \eta$ has leading exponent $\rho +\sigma$. This follows by multiplying the leading terms in Cantor normal form and using continuity of ordinal multiplication at limit ordinals.