"Most of the time" is somewhat problematic, since we are dealing with infinite sets of numbers. But it is tempting to say that. The fact is, any number can be a cube root, since you can cube any number by multiplying it by itself twice. But the cube root of a whole number is always either a whole number or an irrational number. And it is true that if N is any reasonably large whole number (say, 1000 or more), the majority (in fact, at least 99%) of the whole numbers from 1 to N have irrational cube roots.
Answer 1
No. Most of the time it's an irrational number.
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any number that doesn't have a cube root eg (33) 27 is a cubic number cube root is 3
No, it may be not even a rational number. Square root of 2 is 1,414213562... for example.
Any odd root, for example the cube root, of -3.24 is a real number.
6859. Cube root of 9999 is 21.54, so find the largest prime number less than that (19), then cube that number. 19^3=6,859
1728 is even so it cannot have an odd cube root.