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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 $b^5$?

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    Above holds for n>12012-06-02

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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

$(\eta+a)^n=\eta^n+\eta^{n-1}\cdot a+\eta^{n-2}\cdot a+\ldots+\eta^2\cdot a+\eta\cdot a+a\;.$

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    That's exactl$y$ how i proved i$t$. Thanks :)2012-06-02