To express \( 2^{-1 \frac{1}{2}} \) as a root, we first convert the mixed number to an improper fraction:
\[ -1 \frac{1}{2} = -\frac{3}{2} \]
Now, we can rewrite \( 2^{-1 \frac{1}{2}} \) as \( 2^{-\frac{3}{2}} \).
Using the properties of exponents, we have:
\[ 2^{-\frac{3}{2}} = \frac{1}{2^{\frac{3}{2}}} \]
Next, we can express \( 2^{\frac{3}{2}} \) in terms of a root:
\[ 2^{\frac{3}{2}} = \sqrt{2^3} = \sqrt{8} \]
Therefore, we can express \( 2^{-1 \frac{1}{2}} \) as:
\[ 2^{-1 \frac{1}{2}} = \frac{1}{\sqrt{8}} \]
or alternatively, because \( \sqrt{8} = 2\sqrt{2} \):
\[ 2^{-1 \frac{1}{2}} = \frac{1}{2\sqrt{2}} \]
Thus, in root form, \( 2^{-1 \frac{1}{2}} \) can be expressed as \( \frac{1}{\sqrt{8}} \).
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