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

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SUMMARY

The equation x3 = -0.5 has three solutions: one real number and two complex numbers. The real solution is approximately -0.7937, while the complex solutions are approximately 0.3968 + 0.6874i and 0.3969 - 0.6974i. The discussion highlights the distinction between real and complex numbers, particularly in how square roots are defined in the real number system versus the complex number system.

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phymatter
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what is (-1)1/2
 
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There are two square roots of -1, denoted +i and -i.
 


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
 


phymatter said:
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.
 


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.
 
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