To solve the equation \( x^8 = -8 \), we notice that the left side, \( x^8 \), represents any real number raised to an even power, which means it can never be negative.
Thus, there are no real solutions to this equation because \( x^8 \) can only take on non-negative values (0 or positive).
If we were to consider complex solutions, we would look for \( x \) such that:
\[ x^8 = -8 \]
This can be rewritten as:
\[ x^8 = 8e^{i\pi} \]
Taking the eighth root of both sides would give us:
\[ x = 8^{1/8} e^{i(\pi + 2k\pi)/8}, \quad k = 0, 1, 2, \ldots, 7 \]
However, in the context of real numbers, the solutions to \( x^8 = -8 \) do not exist. Therefore, the correct response for real solutions is that there are no solutions. If you need to select from the given responses, you would mark that none of the options are correct for real \( x \).