Applet: Zero to the power of zero is undefined
The color of the point $P$ represents the value of $x^a$, where $x$ is the horizontal coordinate of $P$ (read from the $x$-axis) and $a$ is the vertical coordinate of $P$ (read from the $y$-axis). Red corresponds to $x^a=1$ and blue corresponds to $x^a=0$. You can change both $x$ and $a$ by dragging the point $P$ with your mouse or change one of the values by dragging the corresponding green point on the sliders. The equations in the upper right corner show the current values based on the position of $P$. The point $P$ leaves a trail of colored dots indicating its previous values. Along the $x$-axis, the fact that $a=0$ makes $x^a=1$, and the point $P$ is red. Along $y$-axis, the fact that $x=0$ makes $x^a=0$, and the point $P$ is blue. When both $x=0$ and $a=0$, the value of $x^a$ is undefined, and the point $P$ is yellow. (It's easiest to make both zero using the green points on the sliders.) Since both red points and blue points occur arbitrarily close to the yellow point, we cannot argue that $0^0$ should be either red ($0^0=1$) or blue ($0^0=0$) so we are forced to leave it undefined.
Applet file: zero_to_power_of_zero_undefined.ggb
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