To rewrite the expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] in its simplest radical form, follow these steps:
1. Rewrite the exponent: Notice that the exponent [tex]\(-\frac{3}{6}\)[/tex] can be simplified. The fraction [tex]\(-\frac{3}{6}\)[/tex] simplifies to [tex]\(-\frac{1}{2}\)[/tex]:
[tex]\[
x^{-\frac{3}{6}} = x^{-\frac{1}{2}}
\][/tex]
2. Interpret the negative exponent: To simplify an expression with a negative exponent, recall that [tex]\(x^{-a} = \frac{1}{x^a}\)[/tex]. Thus:
[tex]\[
x^{-\frac{1}{2}} = \frac{1}{x^{\frac{1}{2}}}
\][/tex]
3. Substitute the simplified exponent: Substitute [tex]\(x^{-\frac{1}{2}}\)[/tex] back into the original expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex]:
[tex]\[
\frac{1}{x^{-\frac{1}{2}}} = \frac{1}{\frac{1}{x^{\frac{1}{2}}}}
\][/tex]
4. Reciprocal of the fraction: The reciprocal of [tex]\(\frac{1}{x^{\frac{1}{2}}}\)[/tex] is [tex]\(x^{\frac{1}{2}}\)[/tex]. Therefore:
[tex]\[
\frac{1}{\frac{1}{x^{\frac{1}{2}}}} = x^{\frac{1}{2}}
\][/tex]
5. Rewrite in radical form: The expression [tex]\(x^{\frac{1}{2}}\)[/tex] can be written in its simplest radical form as [tex]\(\sqrt{x}\)[/tex]:
[tex]\[
x^{\frac{1}{2}} = \sqrt{x}
\][/tex]
Therefore, the simplest radical form of [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] is [tex]\(\sqrt{x}\)[/tex].
