Consider doing exact arithmetic on expressions consisting of integers + - * / sqrt. (If we omit sqrt, it's rational number arithmetic, a well understood domain. Inspired by rational number arithmetic libraries that Just Work.)

Difficult operations: Is an expression less than, equal to, or greater than zero? Is an expression real? floor(x).

If we are willing to suffer terrible computational complexity, what are some algorithms?

Does the problem get harder if we permit arbitrary rational powers beyond 1/2?

Resources: sqrtsum, thesis ("Simplifying expressions involving radicals") and papers by Johannes Bloemer, https://math.stackexchange.com/questions/4680/simplification-of-expressions-containing-radicals

Instead of square root, what if the one additional function is logarithm?

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