Cube root If "x" and "y" are allowed to be complex, then there are three solutions (if "x" is non-zero) and so "x" has three cube roots. A real number has one real cube root and two further cube roots which form a complex conjugate pair. For instance, the cube roots of 1 are: The last two of these roots lead to a relationship between all roots of any real or complex number. If a number is one cube root of a particular real or complex number, the other two cube roots can be found by multiplying that cube root by one
Doubling the cube was the discovery by Hippocrates of Chios that it is equivalent to finding two mean proportionals between a line segment and another with twice the length. In modern notation, this means that given segments of lengths and , the duplication of the cube is equivalent to finding segments of lengths and so that In turn, this means that But Pierre Wantzel proved in 1837 that the cube root of 2 is not constructible; that is, it cannot be constructed with straightedge and compass. Menaechmus' original solution involves the intersection of two conic curves. Other more complicated methods of doubling the