Simplifying products of fractional powers of powers

TL;DR

This paper identifies common inaccuracies in computer algebra systems when simplifying products of fractional powers, demonstrates their flaws through extensive testing, and proposes straightforward fixes to improve correctness.

Contribution

It highlights specific simplification errors in popular systems and provides solutions to correct the handling of products of fractional powers.

Findings

86 examples tested across five systems

11% results not equivalent to input

50% results did not simplify to 0 correctly

Abstract

Most computer algebra systems incorrectly simplify (z - z)/(sqrt(w^2)/w^3 - 1/(w*sqrt(w^2))) to 0 rather than to 0/0. The reasons for this are: 1. The default simplification doesn't succeed in simplifying the denominator to 0. 2. There is a rule that 0 is the result of 0 divided by anything that doesn't simplify to either 0 or 0/0. Try it on your computer algebra systems! This article describes how to simplify products of the form w^a*(w^b1)^g1 ... (w^bn)^gn 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 for n=1 on Derive, Maple, Mathematica, Maxima and TI-CAS. Totaled over all five systems, 11% of the results were not equivalent to the input everywhere, 50% of the results did not simplify to 0 a result that was equivalent to 0, and at least 16% of the results exhibited one or more of four additional flaw types. There was substantial room for improvement in all five systems, including the two for which I was a co-author. The good news is: These flaws are easy to fix.