lowest terms of a rational expression

A rational expression is in lowest terms when its numerator and denominator share no common factor other than 1 — there is simply nothing left to cancel. It is the polynomial echo of writing a number fraction as 3/4 rather than 6/8: the same value, expressed as cleanly as possible.

To reach lowest terms you factor the top and bottom completely and cancel every factor they have in common; when no shared factor remains, you are done. The test is not whether the expression looks short, but whether the greatest common factor of numerator and denominator is 1.

An expression in lowest terms is the standard, canonical way to present a rational expression, and it makes later work (adding, comparing, solving) easier. But remember the recurring caveat: reducing to lowest terms can hide an excluded value that a cancelled factor used to enforce, so the original domain restrictions still apply even though the tidy final form no longer displays them.