In this video, I'll show how one can find pairs of numbers that can be commuted under exponentiation. That is, we can find pairs of numbers such that x^y = y^x. We will take this equation, x^y = y^x and parametrize it to find these (x,y) pairs. It turns out that there are infinitely many nice rational number pairs that will commute.
Ideas were adapted from the following paper:
http://www.ww.amc12.org/sites/default/files/pdf/upload_library/22/Chauvenet/Knoebelchv.pdf