To rewrite the expression
[tex]\[
\frac{1}{x^{-\frac{3}{6}}}
\][/tex]
in its simplest radical form, we should simplify and manipulate the expression step-by-step.
### Step 1: Simplify the Exponent
The exponent [tex]\(-\frac{3}{6}\)[/tex] can be simplified.
Simplifying the fraction [tex]\(-\frac{3}{6}\)[/tex] we get:
[tex]\[
-\frac{3}{6} = -\frac{1}{2}
\][/tex]
Thus,
[tex]\[
x^{-\frac{3}{6}} = x^{-\frac{1}{2}}
\][/tex]
### Step 2: Invert the Expression using the Negative Exponent Rule
Recall the rule that states [tex]\( x^{-a} = \frac{1}{x^a} \)[/tex]. Applying this rule, we get:
[tex]\[
x^{-\frac{1}{2}} = \frac{1}{x^{\frac{1}{2}}}
\][/tex]
### Step 3: Substitute back into the original expression
Substituting [tex]\( x^{-\frac{1}{2}} \)[/tex] into the original fraction, we have:
[tex]\[
\frac{1}{x^{-\frac{1}{2}}} = \frac{1}{ \frac{1}{x^{\frac{1}{2}}} }
\][/tex]
### Step 4: Simplify the Fraction
Simplifying the compound fraction [tex]\(\frac{1}{ \frac{1}{x^{\frac{1}{2}}} }\)[/tex], we get:
[tex]\[
\frac{1}{ \frac{1}{x^{\frac{1}{2}}} } = x^{\frac{1}{2}}
\][/tex]
### Step 5: Convert to Radical Form
The exponent [tex]\(\frac{1}{2}\)[/tex] can be rewritten in radical form. Recall that:
[tex]\[
x^{\frac{1}{2}} = \sqrt{x}
\][/tex]
Thus,
[tex]\[
x^{\frac{1}{2}} = \sqrt{x}
\][/tex]
### Conclusion
So, the simplest radical form of the expression
[tex]\[
\frac{1}{x^{-\frac{3}{6}}}
\][/tex]
is
[tex]\[
\sqrt{x}
\][/tex]
