To solve the equation \( m^8 - 4 = -12 \), we first isolate \( m^8 \):
\[ m^8 - 4 + 4 = -12 + 4 \]
This simplifies to:
\[ m^8 = -8 \]
Next, to find \( m \), we take the eighth root of both sides:
\[ m = \sqrt[8]{-8} \]
Since the eighth root of a negative number is not a real number, we can express \(-8\) as:
\[ -8 = 8 \cdot (-1) = 2^3 \cdot (-1) \]
So, we can rewrite:
\[ m^8 = 8 \cdot (-1) \]
Analyzing further: \[ m = \sqrt[8]{2^3 \cdot (-1)} = \sqrt[8]{2^3} \cdot \sqrt[8]{-1} \]
The eighth root of \(-1\) is a complex number, specifically:
\[ \sqrt[8]{-1} = e^{i(\pi/8 + k\pi/4)}, \quad k = 0, 1, 2, 3, 4, 5, 6, 7 \]
But given the options you provided (which seem like integers), it looks like we might be looking for a solution using real numbers.
Looking over the choices:
- m = -64
- m = -1
- m = -128
- m = -2
None of these yield a real number solution since \(m^8 = -8\) has no real solutions. Therefore, if the context strictly relates to integer values, no solution in the provided options is correct.
If you are open to complex values, \( m \) can take on a complex value, but none of the provided options are valid as solutions to the equation given.
If we just substitute back into the original equation for any of the provided options, none of them satisfy:
Thus, it seems likely there is no solution among the options provided.