Question

How to find the complex cube roots of -1?

Answer 1

-1=i^2
so complex number cube root would be
\[\sqrt[3]{i ^{2}}\]
\[(i ^{2})^{1/3}\]
\[i ^{2/3}\]
\[i ^{2}-i ^{3}\]
\[-1-\sqrt{-1}\]
but the final answer would just be until i^2/3 i think since it is asked in terms of complex numbers

Answer 2

It may be simpler to do as follows:
\[x^3= -1 \rightarrow x^3+1=0 \rightarrow (x+1)(x^2-x+1)=0\]
This gives x = -1 as the real root and the two complex roots can be obtained by solving the quadratic equation\[x^2-x+1\]