Factoring $a^p \pm b^p$

Question

I know that if $p$ is a positive integer then $a^p \pm b^p$ can be factored using the well-known formulas(note : $a^p+b^p$ can be factored this way only when $p$ is odd). Now, what if $p$ is, for instance, a rational number? But what if it is an irrational number? Are there similar ways to express $a^p \pm b^p$?

Answer 1

The first question you need to ask, if $p$ is not an integer, is what do you even mean by "factoring" this expression.

But if you insist, for any positive integer $n$ you can write $a^p = x^n$ and $b^p = y^n$ where $x = a^{p/n}$ and $y = b^{p/n}$ (with appropriate caveats in case $a$ or $b$ is negative), and then factor $x^n \pm y^n$.