No, the vast majority are not.
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All numbers have cube roots (not necessarily integral cube roots) so every prime has cube roots.
Not sure what answer you are looking for, but here are 4 types of roots in math. First is a square roots, next is cube roots, then the nth roots, and lastly rational roots.
No. The square roots of perfect squares are rational.
Not necessarily. The cube roots of 4, 6 and 9 are all irrational (and different). But their product is 6, not just rational, but an integer.
Yes.