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

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We define i= \sqrt{-1}.This conversation discusses the possibility of a positive fractional power of a negative number. The summary is that while there are two square roots of -1, denoted +i and -i, in the real number system, the complex numbers cannot be ordered and therefore do not have a defined positive or negative square root. This means that there are three possible answers to the question of the positive fractional power of a negative number, with one being a real number and the other two being complex numbers.
  • #1
phymatter
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what is (-1)1/2
 
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  • #2


There are two square roots of -1, denoted +i and -i.
 
  • #3


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
 
  • #4


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.
 
  • #5


ONE of the three numbers, x, such that [itex]x^3= -0.5[/itex], 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 [itex]a^{1/2}[/itex] to be the positive number, x, such that [itex]x^2= a[/itex]. 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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FAQ: What are the three solutions to x^3 = -0.5?

What does (-1)1/2 mean?

The notation (-1)1/2 represents the square root of -1, which is a complex number. It is commonly denoted as i, where i2 = -1.

Why does (-1)1/2 equal i?

This is because when a negative number is raised to a fractional exponent, it results in a complex number. In the case of (-1)1/2, the exponent 1/2 represents the square root, and the negative sign indicates that the result is a negative number. Therefore, (-1)1/2 equals i.

Can the square root of a negative number be simplified?

No, the square root of a negative number cannot be simplified further into a real number. It will always result in a complex number, such as i or -i.

How is (-1)1/2 used in mathematics?

The complex number i, represented by (-1)1/2, is used in various branches of mathematics, such as in complex analysis, differential equations, and physics. It is also used to simplify calculations and solve problems that involve imaginary numbers.

Is (-1)1/2 considered a real number?

No, (-1)1/2 is not a real number. It is a complex number, which consists of a real part and an imaginary part. In this case, the real part is 0 and the imaginary part is 1, making it a purely imaginary number.

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