The cube root of -1 ("minus 1" or "negative 1") is -1. In fact all odd-numbered roots (i.e., the fifth root, the seventh root, the ninth root, etc.) of -1 are -1.
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-1X-1X-1=-1 therefore cube root of -1 = -1
The real cube root of 1 is 1, since 13 = 1. There also a pair of complex cube roots.
The volume of a cube is found by cubing the length of an edge. To find the measure of an edge of a cube of a known volume, simply find the cube root of the volume. In this case, the cube root of 1728 is 12. It's actually plus or minus 12, but we drop the minus 12 because shapes can have negative dimensions. Given a cube of known volume, the length of an edge can be found by finding the cube root of the cube's volume.
The cube root of 1 will also be 1. You can check with the complementary operation: 1*1*1=1 (true).