What is the simplification of?
It is proven that a^y=a^ma^n=a^{m-n}=a^aa^b In full concordance with the Law of Indices. As your inquiry thus requires y=z to define a^y=a^z You may as well ease your matters and use the more brief: law of indices. The pro to this is: most are already familiarized by Newton's Concordances. Newton's Concordances are to enable an understanding: Persons and especially merchants and captains who need to deduce a^ya^{-b}=Huge Bulk Quantity=Q=a^a should not constrain small merchants to abuse theyself with a^aa^{-b}=a^y=Qa=aa^{.00001568} When it is 40 bags of grass seed, by: a=grass seeds on an acre a=4000037 or so -b=4000036.99998432 or so Thus comes my long question: As Mr. Newton already conquered this aspect and Major Problem, why should we stray from that to labor beyond: a^ma^n=a^{m-n} To a very laborious, unweilding, and insober process not complicated by only involving a^y but complex by increased likelihood you will need to at some point verify: a^z=a^y=a^aa^{-b}=a^{m-n}=a^{a-(-b)} Which yourself have not yet detailed: you even know how to even convey y=a+b let convert y to this mystery of z, which likely is involving Beal: A^x+B^y=C^z? Explain. Further Reading Dave Palamar's answer to How could we solve 3^x=9x?