the zero-equivalence problem
It sounds like the easiest question in the world: is this expression equal to zero? For 3 - 3, obviously yes. But for exp(log(x)) - x, or for a tangle of nested logs, exponentials, and square roots, deciding whether the whole thing is secretly zero turns out to be one of the hardest problems in computer algebra — and in its full generality, it is provably impossible for a computer to always answer correctly.
The problem is this: given a symbolic expression, decide whether it represents the zero function (equivalently, whether two expressions A and B are equal, by asking if A - B is zero). For polynomials and rational functions it is easy — expand to a canonical form and check if everything cancels. But once you allow the 'elementary' functions (exp, log, sin, radicals, all freely composed), Richardson's theorem shows the question becomes undecidable: there is no algorithm that halts with a correct yes-or-no for every such expression. This is why a perfect canonical form for general expressions cannot exist, and why automatic simplification can be defeated by expressions that are equal but refuse to look equal.
In practice, systems dodge the impossibility rather than solve it. The dominant trick is probabilistic: plug in random numerical values; if the expression comes out clearly non-zero, it is definitely not zero, and if it comes out zero at many random points it is almost certainly the zero function (the Schwartz-Zippel idea behind much of computer algebra). This gives a fast answer that is correct with overwhelming probability but never an absolute proof of zero. The honest takeaway: equality testing in a CAS is a deep, sometimes undecidable problem hiding behind an innocent equals sign, and a 'cannot decide' or a probabilistic verdict is sometimes the most honest answer available.
Is sqrt(3 + 2*sqrt(2)) - 1 - sqrt(2) equal to zero? It is, because 3 + 2*sqrt(2) = (1 + sqrt(2))^2. A CAS may not see this from the surface form, but plugging the expression into a high-precision numerical evaluator gives 0.00000... to fifty digits — strong probabilistic evidence of zero, short of a symbolic proof.
Numerical testing gives strong evidence of zero but not a symbolic proof.
By Richardson's theorem, zero-equivalence for general elementary expressions is undecidable — no algorithm can always answer correctly. Probabilistic tests (random evaluation) are fast and almost always right, but a verdict of 'zero' from them is overwhelming evidence, not a proof.