To rewrite the expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] in simplest radical form, we will go through the following steps:
1. Simplify the exponent [tex]\(-\frac{3}{6}\)[/tex]:
[tex]\[
-\frac{3}{6} = -\frac{1}{2}
\][/tex]
So, we can rewrite the expression as:
[tex]\[
\frac{1}{x^{-\frac{1}{2}}}
\][/tex]
2. Simplify the expression using the property of exponents:
Recall that for any positive number [tex]\(a\)[/tex] and any real number [tex]\(b\)[/tex], [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex]. This means that:
[tex]\[
x^{-\frac{1}{2}} = \frac{1}{x^{\frac{1}{2}}}
\][/tex]
3. Substitute this back into the original expression:
[tex]\[
\frac{1}{\frac{1}{x^{\frac{1}{2}}}}
\][/tex]
4. Simplify the fraction:
Invert the denominator to get rid of the complex fraction:
[tex]\[
\frac{1}{\frac{1}{x^{\frac{1}{2}}}} = x^{\frac{1}{2}}
\][/tex]
5. Convert the exponent form to radical form:
Recall that for any positive number [tex]\(a\)[/tex] and any positive integer [tex]\(n\)[/tex], [tex]\(a^{\frac{1}{n}} = \sqrt[n]{a}\)[/tex]. Hence,
[tex]\[
x^{\frac{1}{2}} = \sqrt{x}
\][/tex]
Therefore, the expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] in the simplest radical form is:
[tex]\[
\boxed{\sqrt{x}}
\][/tex]
