To solve the equation \(x^8 = -8\), we need to analyze the equation.
The left side, \(x^8\), represents the eighth power of \(x\). Since any real number raised to an even power (like 8) is non-negative, \(x^8\) is always greater than or equal to 0 for real numbers.
On the right side, we have \(-8\), which is a negative number.
Since \(x^8\) can never equal a negative number for real \(x\), there are no real solutions to this equation.
However, if we consider complex numbers, we can find solutions.
We can rewrite the equation as:
\[ x^8 = -8 \]
This can be expressed in polar form. We can convert \(-8\) to polar form, which has a magnitude of \(8\) and an angle of \(\pi\) (or \(180^\circ\)). In polar form, we represent \(-8\) as:
\[ 8 (\cos \pi + i \sin \pi) \]
Now we take the eighth root of both sides. The magnitude of the roots will be:
\[ \sqrt[8]{8} = 8^{1/8} = 2^{3/8} \]
And for the angle, we divide by \(8\) and also add multiples of \(\frac{2\pi}{8}\) (which is \(\frac{\pi}{4}\)) to find each of the eight roots.
Thus, the solutions are given by:
\[ x_k = 2^{3/8} \left(\cos \left(\frac{\pi + 2k\pi}{8}\right) + i \sin \left(\frac{\pi + 2k\pi}{8}\right)\right) \quad \text{for } k = 0, 1, 2, \ldots, 7 \]
These roots are complex solutions and not provided as part of your response options.
Since the original question asks for real solutions, there are actually no real solutions to the equation \(x^8 = -8\).
If instead, you want to check the provided options:
- \(x = 64\)
- \(x = 0\)
- \(x = -1\)
- \(x = -64\)
All these options yield \(x^8 \geq 0\), which would not satisfy the equation.
Therefore, based on the context, the correct conclusion is no real solutions.
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