\begin{frame} \frametitle{Exponential Functions: Irrational Numbers} \begin{alertblock}{} But what about irrational numbers? What is $2^{\sqrt{3}}$ or $5^\pi$? \end{alertblock} \bigskip \pause By increasingness we know: \begin{talign} 1.73 < \sqrt{3} < 1.74 &&\implies&& \alert<7->{2^{1.73} < 2^{\sqrt{3}} < 2^{1.74}} \\ \mpause[1]{1.732 < \sqrt{3} < 1.733} &&\mpause[1]{\implies}&& \mpause[1]{\alert<7->{2^{1.732} < 2^{\sqrt{3}} < 2^{1.733}}} \\ \mpause[2]{1.7320 < \sqrt{3} < 1.7321} &&\mpause[2]{\implies}&& \mpause[2]{\alert<7->{2^{1.7320} < 2^{\sqrt{3}} < 2^{1.7321}}} \\ \mpause[3]{1.73205 < \sqrt{3} < 1.73206} &&\mpause[3]{\implies}&& \mpause[3]{\alert<7->{2^{1.73205} < 2^{\sqrt{3}} < 2^{1.73206}}} \\[-1ex] && \mpause[4]{\vdots} && \end{talign} \pause[7]\vspace{-2ex} There is exactly one number that fulfills all \alert{conditions} on the right.\hspace*{-.5cm} \pause\bigskip E.g., $2^{1.73205} < 2^{\sqrt{3}} < 2^{1.73206}$ determines the first $6$ digits: \begin{talign} 2^{\sqrt{3}} \approx 3.321997 \end{talign} \vspace{10cm} \end{frame}