Ordinal Exponentiation via Cantor Normal Form/Corollary

Theorem
Let $x$ and $y$ be ordinals.

Let $x$ be a limit ordinal and let $y > 0$.

Let $\langle a_i \rangle$ be a sequence of ordinals that is strictly decreasing on $1 \le i \le n$.

Let $\langle b_i \rangle$ be a sequence of natural numbers.

Then:


 * $\displaystyle \left({ \sum_{i \mathop = 1}^n x^{a_i} \times b_i }\right)^{x^y} = x^{a_1 \mathop \times x^y}$

Proof
By the hypothesis, $x^y$ is a limit ordinal by Limit Ordinals Preserved Under Ordinal Exponentiation.

The result follows from Ordinal Exponentiation via Cantor Normal Form/Limit Exponents.

Also see

 * Cantor Normal Form