Finite or infinite number of roots!

Algebra Level 4

The equation \(x^3=-1\) has three solutions, one of which is real and the other two are non-real complex numbers. Determine the number and type of solutions of \[ \large x^{\frac{1}{\sqrt{2}}}=-1\]

Note: When \(x\) is a complex number different from \(0\), and \(r\) is a real number, \(x^r\) can have more than one possible value. In this case, we assume that the complex number \(x\) is a solution of the equation \(x^r=s,\) where \(s\) is a given real number, if at least one of the values of \(x^r\) is equal to \(s.\)

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