Question

Solve for x . x8=−8 (1 point) Responses x=64 x equals 64 x=0 x equals 0 x=−1 x equals negative 1 x=−64

Answer 1

To solve the equation \( x^8 = -8 \), we need to determine if there are any real solutions.

Notice that the left side, \( x^8 \), represents a number raised to an even power, which means it can never be negative. The right side of the equation, \(-8\), is negative. Since a non-negative number cannot equal a negative number, there are no real solutions for this equation.

We can still consider complex solutions. If we take the eighth root of both sides, we can express:

\[
x = (-8)^{1/8}
\]

Expressing \(-8\) in polar form, we have:

\[
-8 = 8(\cos \pi + i\sin \pi)
\]

From this, the eighth roots can be found using:

\[
x = 2\left( \cos\left(\frac{\pi + 2k\pi}{8}\right) + i\sin\left(\frac{\pi + 2k\pi}{8}\right) \right) \quad \text{for } k = 0, 1, 2, \ldots, 7
\]

These roots provide complex solutions.

However, within the context of the responses given (64, 0, -1, -64), none of these options are valid solutions to the equation \( x^8 = -8 \).

So, the correct conclusion is:
**There are no real solutions to the equation \( x^8 = -8 \).**