To solve the equation \( x^8 = -8 \), we need to consider that for any real number \( x \), \( x^8 \) is non-negative (since raising a number to an even power results in a non-negative number). Therefore, there are no real solutions for \( x^8 = -8 \).
If we're only considering real numbers, the correct conclusion is that there is no solution. However, if the question allows for complex solutions, we can proceed differently.
The solutions can be found using complex numbers, and \( x^8 = -8 \) can be rewritten in polar form.
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We express \(-8\) in polar coordinates. It has a modulus of \(8\) and an argument of \(\pi\) radians (or \(180^\circ\)).
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Therefore, we can set: \[ x^8 = 8(\cos(\pi) + i\sin(\pi)) \]
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We can apply the \(8^{\text{th}}\) root: \[ x = 8^{1/8} \text{cis}\left(\frac{\pi + 2k\pi}{8}\right) \] Where \( k = 0, 1, 2, ..., 7 \).
Calculating \(8^{1/8}\): \[ 8^{1/8} = 2^{3/8} = 2^{0.375} \]
Finding the angles for \(k = 0\) to \(7\):
- \( k = 0 \Rightarrow x = 2^{0.375}\text{cis}\left(\frac{\pi}{8}\right) \)
- \( k = 1 \Rightarrow x = 2^{0.375}\text{cis}\left(\frac{3\pi}{8}\right) \)
- \( k = 2 \Rightarrow x = 2^{0.375}\text{cis}\left(\frac{5\pi}{8}\right) \)
- \( k = 3 \Rightarrow x = 2^{0.375}\text{cis}\left(\frac{7\pi}{8}\right) \)
- \( k = 4 \Rightarrow x = 2^{0.375}\text{cis}\left(\frac{9\pi}{8}\right) \)
- \( k = 5 \Rightarrow x = 2^{0.375}\text{cis}\left(\frac{11\pi}{8}\right) \)
- \( k = 6 \Rightarrow x = 2^{0.375}\text{cis}\left(\frac{13\pi}{8}\right) \)
- \( k = 7 \Rightarrow x = 2^{0.375}\text{cis}\left(\frac{15\pi}{8}\right) \)
Since the options provided do not match any results from above or any imaginary solutions, based on the original question context, I will reaffirm that the correct answer is that there is no real solution.
Thus, if forced to choose from the provided responses, none are correct, as \( x^8 = -8 \) lacks real solutions. If seeking complex solutions, then we would determine them as stated.
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