The problem is to simplify the expression $\frac{1}{2\sqrt{x^{-8}}}$.

1. Problem Description

The problem is to simplify the expression $\frac{1}{2\sqrt{x^{-8}}}$.

2. Solution Steps

We start with the expression $\frac{1}{2\sqrt{x^{-8}}}$.

We can rewrite the expression as:

$\frac{1}{2(x^{-8})^{\frac{1}{2}}}$

Using the property $(a^m)^n = a^{m \times n}$, we can simplify the exponent:

$\frac{1}{2x^{-8 \times \frac{1}{2}}}$

$\frac{1}{2x^{-4}}$

Using the property $a^{-n} = \frac{1}{a^n}$, we can rewrite $x^{-4}$ as $\frac{1}{x^4}$.

$\frac{1}{2 \cdot \frac{1}{x^4}}$

We can rewrite the expression as:

$\frac{1}{\frac{2}{x^4}}$

Dividing by a fraction is the same as multiplying by its reciprocal:

$1 \cdot \frac{x^4}{2}$

Thus, the simplified expression is:

$\frac{x^4}{2}$

3. Final Answer

$\frac{x^4}{2}$