To rewrite the expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] in its simplest radical form, follow these steps:
1. Simplify the exponent:
[tex]\[
-\frac{3}{6} = -\frac{1}{2}
\][/tex]
Therefore, the expression becomes:
[tex]\[
\frac{1}{x^{-\frac{1}{2}}}
\][/tex]
2. Rewrite using exponential rules:
Recall that [tex]\(x^{-a} = \frac{1}{x^a}\)[/tex]. So, [tex]\(x^{-\frac{1}{2}}\)[/tex] is:
[tex]\[
\frac{1}{x^{-\frac{1}{2}}} = x^{\frac{1}{2}}
\][/tex]
3. Convert to radical form:
The exponent [tex]\(\frac{1}{2}\)[/tex] signifies the square root. So, [tex]\(x^{\frac{1}{2}}\)[/tex] can be written as:
[tex]\[
\sqrt{x}
\][/tex]
Now, placing it all together, the original expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] in simplest radical form is:
[tex]\[
\boxed{\sqrt{x}}
\][/tex]
The step-by-step process yields the final simplified radical form of the given expression as [tex]\(\sqrt{x}\)[/tex].
