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

AlgebraSimplificationExponentsRadicals
2025/3/8

1. Problem Description

The problem is to simplify the expression 12x8\frac{1}{2\sqrt{x^{-8}}}.

2. Solution Steps

We start with the expression 12x8\frac{1}{2\sqrt{x^{-8}}}.
We can rewrite the expression as:
12(x8)12\frac{1}{2(x^{-8})^{\frac{1}{2}}}
Using the property (am)n=am×n(a^m)^n = a^{m \times n}, we can simplify the exponent:
12x8×12\frac{1}{2x^{-8 \times \frac{1}{2}}}
12x4\frac{1}{2x^{-4}}
Using the property an=1ana^{-n} = \frac{1}{a^n}, we can rewrite x4x^{-4} as 1x4\frac{1}{x^4}.
121x4\frac{1}{2 \cdot \frac{1}{x^4}}
We can rewrite the expression as:
12x4\frac{1}{\frac{2}{x^4}}
Dividing by a fraction is the same as multiplying by its reciprocal:
1x421 \cdot \frac{x^4}{2}
Thus, the simplified expression is:
x42\frac{x^4}{2}

3. Final Answer

x42\frac{x^4}{2}

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