What are the three solutions to x^3 = -0.5?

[phymatter]

what is (-1)^1/2

 

[CRGreathouse]

There are two square roots of -1, denoted +i and -i.

 

[phymatter]

actually i missed the point , i wanted to ask that is the positive fractional power of a negative number possible , like (-0.5)^1/3

 

[CRGreathouse]

actually i missed the point , i wanted to ask that is the positive fractional power of a negative number possible , like (-0.5)^1/3

The point is that there are two answers to your first question and three to your second.

 

[HallsofIvy]

ONE of the three numbers, x, such that $x^3= -0.5$, is a real number (it is approximately -0.7937), the other are two complex numbers, 0.3968+ 0.6874i and 0.3969- 0.6974i, approximately (and assuming I have done the arithmetic correctly).

In the real number system we can distinguish between two square roots in that one is positive and the other negative. And we define $a^{1/2}$ to be the positive number, x, such that $x^2= a$. The complex numbers, however, cannot be made into an ordered field so there is no way of distinguishing one of the two square roots.