This is a proof of the Power Rule. I wrote this during a vodcast I created which is on YouTube at my channel, which is <a href="https://www.youtube.com/user/oemb1905z/channels" target="_blank">YouTube</a>.
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Although the simplest rule to execute algebraically, it is the most complex to prove rigorously. I also included a version of Pascal's triangle that I created to help explain the proof to my students, which is included above the proof itself.