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The power operator has higher priority than the unary negation.

So in the first case you are really doing "minus (1 to the 1/3)", and not "(minus 1) to the 1/3".

For the real n-th root, use real_nth_root.

sage: real_nth_root(-1, 3)
-1

Regarding cubing the cube root:

sage: a = (-1)^(1/3)
sage: a
(-1)^(1/3)
sage: a^3
-1

Using the numerical approximation of the cube root:

sage: aa = a.n()
sage: aa
0.500000000000000 + 0.866025403784439*I
sage: aa^3
-1.00000000000000 + 3.88578058618805e-16*I

So, very close to -1, with tiny imaginary part from rounding errors.