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\begin{frame}
  \frametitle{Exponential Functions: Irrational Numbers}
  
  \begin{alertblock}{}
    But what about irrational numbers?
    What is $2^{\sqrt{3}}$ or $5^\pi$?
  \end{alertblock}
  \pause\bigskip
  
  Roughly, one can imagine the situation like in this figure:
  \begin{center}
  \scalebox{.7}{
  \begin{tikzpicture}[default,baseline=0cm]
    \diagram{-2.5}{2.5}{-1}{4}{1}
    \diagramannotatez
    \diagramannotatexx{-1,1}
    \diagramannotatey{1}
    \begin{scope}[ultra thick]
    \draw[cgreen,dotted] plot[smooth,domain=-2.5:2,samples=300] (\x,{2^\x});
    \end{scope}
  \end{tikzpicture}
  }
  \end{center}
  \pause
  
  We have have defined the function for all rational points, and
  now want to close the gaps.
  \pause
  \bigskip
  
  Clearly, the result should be an increasing function\ldots
  \vspace{10cm}
\end{frame}