To solve the equation \( x^8 = -8 \), we need to recognize that \( x^8 \) (the eighth power of \( x \)) is always non-negative for any real number \( x \). Therefore, there is no real solution to this equation, as \( -8 \) is negative.
If we consider complex numbers, we can express the solution as follows:
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Rewrite the equation as: \[ x^8 = -8 \]
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Express \(-8\) in polar form: \[ -8 = 8(\cos(\pi) + i\sin(\pi)) \]
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The eighth roots can be found using De Moivre's theorem: \[ x = \sqrt[8]{8} \left( \cos\left(\frac{\pi + 2k\pi}{8}\right) + i\sin\left(\frac{\pi + 2k\pi}{8}\right) \right) \quad (k = 0, 1, 2, \ldots, 7) \]
Given that we’re specifically looking for the real number solutions, we conclude that there is no appropriate \( x \) that satisfies the equation \( x^8 = -8 \) among the provided options.
Hence, the answer is that none of the options provided is correct.