The complex number x+iy has two alternative forms, the polar form r(cos(theta)+i sin(theta)) and the exponential form r.e^(i theta), where r is the positive square root of x^2 + y^2 and theta is arctan(y/x). Both polar and exponential form are often abbreviated to r,angle(theta). (Apologies for the text by the way, I can't be bothered fiddling around with LaTex.)

The angle theta is multivalued and strictly should be written theta + 2k.pi, where k is any integer, positive or negative. In most cases the 2k.pi is omitted, but, if the complex number is being raised to a fractional power, it must be included.

Consider e^{i(theta+2k.pi)} raised to the power one third. Using the usual rule for indices this becomes e^{i(theta/3 + 2k.pi/3)} and, as you can see, the angle takes on different values for different values of k. In fact, ignoring the multivaluedness, there will be just three different angles, theta/3, theta/3+2pi/3 and theta/3+4pi/3. The next one theta/3+6pi/3 = theta/3+2pi is a repeat of theta/3. Other values of k simply repeat one of these three angles.

So, there will be three different cube roots of a number, and this applies to any number, not just complex ones. For example, there are three cube roots of 8, 2 angle 0 = 2, 2 angle 2pi/3 = 2(-1/2 + i 3^(1/2)/2) = -1 + i.3^(1/2) and 2 angle 4pi/3 = -1 - i.3^(1/2).

Similarly there will be two square roots (which everyone is used to for positive numbers at least), four fourth roots, five fifth roots and so on.