Simplifying products of fractional powers of powers

Abstract

Most computer algebra systems incorrectly simplify z -- z /√ w 2 1/ w 3 -- w √ w 2 to 0 rather than to 0/0. The reasons for this are: The default simplification does not succeed in simplifying the denominator to 0. There is a rule that 0 is the result of 0 divided by anything that does not simplify to either 0 or 0/0. . Many of these systems have more powerful optional transformation and general purpose simplification functions. However that is unlikely to help this example even if one of those functions can simplify the denominator to 0, because the input to those functions is the result of default simplification, which has already incorrectly simplified the overall ratio to 0. Try it on your computer algebra systems! This article describes how to simplify products of the form w α ( w β1 ) λ1 ... ( w β n ) λ n correctly and well, where w is any real or complex expression and the exponents are rational numbers. It might seem that correct good simplification of such a restrictive expression class must already be published and/or built into at least one widely used computer-algebra system, but apparently this issue has been overlooked. Default and relevant optional simplification was tested with 86 examples on 5 systems with n = 1. Using a spectrum from the most serious flaw being a result that is not equivalent to the input somewhere to the least serious being not rationalizing a denominator when that does not cause a more serious flaw, the overall percentage of most flaw types is alarming.