Nonlimit ordinal exponentiation I know how to compute finite exponentiation limit ordinal expressed in normal form.
However, how do i compute for nonlimit ordinal?
I know this statement is true.
If $r$ is a limit ordinal and $b$ is a finite ordinal >0, then $r^b = \omega^{a_1b-1}r$
How do i compute ordinal such as
$\omega^\omega + 12 = b$ then $b^5$?
 A: For finite $a$ you have 
$$\begin{align*}
\left(\omega^\omega+a\right)^2&=\left(\omega^\omega+a\right)\cdot\left(\omega^\omega+a\right)\\
&=\left(\omega^\omega+a\right)\cdot\omega^\omega+\left(\omega^\omega+a\right)\cdot a\\
&=\omega^{\omega+\omega}+\omega^\omega\cdot a+a\;,
\end{align*}$$
$$\begin{align*}
\left(\omega^\omega+a\right)^3&=\left(\omega^{\omega+\omega}+\omega^\omega\cdot a+a\right)\cdot\left(\omega^\omega+a\right)\\
&=\left(\omega^{\omega+\omega}+\omega^\omega\cdot a+a\right)\cdot\omega^\omega+\left(\omega^{\omega+\omega}+\omega^\omega\cdot a+a\right)\cdot a\\
&=\omega^{\omega\cdot3}+\left(\omega^{\omega\cdot2}\cdot a+\omega^\omega\cdot a+a\right)\\
&=\omega^{\omega\cdot3}+\omega^{\omega\cdot2}\cdot a+\omega^\omega\cdot a+a\;,
\end{align*}$$
and
$$\begin{align*}
\left(\omega^\omega+a\right)^4&=\left(\omega^{\omega\cdot3}+\omega^{\omega\cdot2}\cdot a+\omega^\omega\cdot a+a\right)\cdot\left(\omega^\omega+a\right)\\
&=\left(\omega^{\omega\cdot3}+\omega^{\omega\cdot2}\cdot a+\omega^\omega\cdot a+a\right)\cdot\omega^\omega+\left(\omega^{\omega\cdot3}+\omega^{\omega\cdot2}\cdot a+\omega^\omega\cdot a+a\right)\cdot a\\
&=\omega^{\omega\cdot4}+\left(\omega^{\omega\cdot3}\cdot a+\omega^{\omega\cdot2}\cdot a+\omega^\omega\cdot a+a\right)\\
&=\omega^{\omega\cdot4}+\omega^{\omega\cdot3}\cdot a+\omega^{\omega\cdot2}\cdot a+\omega^\omega\cdot a+a\;,
\end{align*}$$
and the induction is pretty clear:
$$\left(\omega^\omega+a\right)^n=\omega^{\omega\cdot n}+\omega^{\omega\cdot(n-1)}\cdot a+\omega^{\omega\cdot(n-2)}\cdot a+\ldots+\omega^{\omega\cdot2}\cdot a+\omega^\omega\cdot a+a\;.$$
In particular,
$$\left(\omega^\omega+12\right)^5=\omega^{\omega\cdot5}+\omega^{\omega\cdot4}\cdot12+\omega^{\omega\cdot3}\cdot12+\omega^{\omega\cdot2}\cdot12+\omega^\omega\cdot12+12\;.$$
In general, if $\eta$ is a limit ordinal, you’ll have
$$(\eta+a)^n=\eta^n+\eta^{n-1}\cdot a+\eta^{n-2}\cdot a+\ldots+\eta^2\cdot a+\eta\cdot a+a\;.$$
