\begin{frame}
  \frametitle{Exponential Functions}

  \begin{exampleblock}{}
  How is $a^x$ defined for $x \in \mathbb{R}$?
  \end{exampleblock}
  \bigskip\pause
  
  For $x = 0$ we have $a^0 = 1$.
  \pause\bigskip

  For positive integers $x = n \in \nat$ we have
  \begin{talign}
    a^n = \underbrace{a\cdot a \cdot \cdots \cdot a}_{\text{$n$-times}}
  \end{talign} 
  \pause\vspace{-1ex}
  
  For negative integers $x = -n$ we have
  \begin{talign}
    a^{-n} = \frac{1}{a^n}
  \end{talign} 
  \pause\vspace{-1.5ex}
  
  For rational numbers $x = \frac{p}{q}$ with $p,q$ integers we have
  \begin{talign}
    a^x = a^{\frac{p}{q}} = \sqrt[q]{a^p} \pause = (\sqrt[q]{a})^p
  \end{talign} \vspace{-2ex}
  
  \mpause[0]{
  \begin{exampleblock}{}
    \begin{malign}
      4^{\frac{3}{2}} = \mpause[1]{ (\sqrt[2]{4})^3 =} \mpause[2]{ 2^3 = }\mpause[3]{ 8 }
    \end{malign}
  \end{exampleblock}
  }
\end{frame}