What are the imaginary cube roots of -125?

Answer 1

This can be done easily if you use polar coordinates. I did all of the following calculations in my head, without resorting to a calculator:

One of the cubic roots of -125 is -5. That is the same as 5, at an angle of 180 degrees. The other two cubic roots also have an absolute value of 5, and each cubic root has an angle of 120 degrees to the other cubic roots. In other words, the complex roots are 5 at an angle of 60 degrees, and 5 at an angle of -60 degrees.

If you want to convert this to rectangular coordinates (i.e., show the real and the imaginary parts separately), use the P-->R (polar to rectangular) conversion, available on most scientific calculators.