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
General information about Geogebra Web applets
This applet was created using Geogebra. In most Geogebra applets, you can move objects by dragging them with the mouse. In some, you can enter values with the keyboard. To reset the applet to its original view, click the icon in the upper right hand corner.
You can download the applet onto your own computer so you can use it outside this web page or even modify it to improve it. You simply need to download the above applet file and download the Geogebra program onto your own computer.