Solving Exponential Equation \( 4^{8x} = 256 \)

We need to find the value of $x$ satisfying the equation \( 4^{8x} = 256 \).

1. Rewrite the equation with a common base.

   Recognize that $256$ is a power of $4$. Specifically, $256 = 4^4$. Substituting this into the original equation gives: \( 4^{8x} = 4^4 \)

2. Equate the exponents.

   Since the bases on both sides of the equation are the same ($4$), the exponents must be equal: \( 8x = 4 \)

3. Solve for $x$.

   To isolate $x$, divide both sides of the equation by $8$: \( x = \frac{4}{8} \)

   Simplify the fraction: \( x = \frac{1}{2} \)

The calculation shows that \( x = \frac{1}{2} \). The provided correct answer is Option C ($\frac{2}{3}$).