> None of them had a good answer! These were the top mathematicians in the world.

I can believe that.

Why?

I'm not one of "the top mathematicians in the world", but I do hold a Ph.D. in applied math from a world class research university and have published peer-reviewed research in math.

My view is that (A) the problem, solve for x in x^3 = 27, (B) the request "show your work", and (C) the objective of the work of the OP to "simplify" some algebraic expressions are at best flawed introductions to math as pure/applied mathematicians do it and in our educational system as efforts in, call it, pedagogy.

IMHO the goal of "simplify" an algebraic expression is especially flawed; that began to dawn on me in high school, and I so concluded in college and since.

Why? To "simplify" an algebraic expression is mostly a matter of style often without clear criteria or a unique answer; such simplification can at times be to illustrate something particular to the context but not be general.

Really, in math, we manipulate/leave algebraic expressions in whatever form is useful for what we are doing with the expressions and, IMHO, essentially never much for a goal of mere style or simplification.

E.g., an important manipulation of algebraic expressions was taking the algebra for discrete Fourier transforms and, essentially, manipulating it to illustrate how to do the calculations of the fast Fourier transform (FFT) -- work mostly of J. Tukey, supposedly at a US Presidential Science Advisers meeting to answer a question of R. Garwin. The FFT is darned important, and curiously the main point can be discovered and illustrated just by manipulating the algebraic expression to be in one of several particular forms.

Too much in math as commonly taught in K-12 and early college isn't really close to math as done by people really using math in, say, the STEM fields but is stuffed in there by the teachers as part of pedagogy or having a source of exercises and test questions.

In response, generally it would be good to lower the emphasis on such make work pedagogy, get the students through it (minimize it and have lenient grading of it), and get on to what is important in math and its applications, research, etc.

E.g., currently a big problem and hot topic in applied/research math is over fitting. Well, hush, don't tell anyone, but in some important cases can make some surprisingly good progress on over-fitting, realliy, get rid of the concerns, by essentially rewriting some of the algebra and just looking and observing. How 'bout that! No, I don't offer to fill in the details! Uh, in some cases, this work can also be a great way around some really nasty numerical stability problems.

But, right, simplifying some algebra can be important when have an important objective in mine, and style is not such an objective and, really, is not a good guide to what would be a simplification useful for some important objective.

Or, with the FFT and over-fitting, I've given two cases where there is an important objective for simplifying an algebraic expression -- alas, in both cases, without the important objective in mind, neither simplification would be seen to have better style!