Sure! Let's simplify the expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] step-by-step.
1. Simplify the exponent: The exponent [tex]\(-\frac{3}{6}\)[/tex] can be simplified. Since [tex]\(\frac{3}{6} = \frac{1}{2}\)[/tex], the expression becomes:
[tex]\[
-\frac{3}{6} = -\frac{1}{2}
\][/tex]
Therefore, [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] can be rewritten as:
[tex]\[
\frac{1}{x^{-\frac{1}{2}}}
\][/tex]
2. Apply the negative exponent property: The property of exponents states that [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex]. Applying this property to the expression [tex]\(x^{-\frac{1}{2}}\)[/tex], we get:
[tex]\[
x^{-\frac{1}{2}} = \frac{1}{x^{\frac{1}{2}}}
\][/tex]
3. Rewrite the initial expression: Using the result from the previous step in our original expression:
[tex]\[
\frac{1}{x^{-\frac{1}{2}}} = x^{\frac{1}{2}}
\][/tex]
4. Express as a radical: The exponent [tex]\(\frac{1}{2}\)[/tex] can be expressed in radical form as the square root. Therefore:
[tex]\[
x^{\frac{1}{2}} = \sqrt{x}
\][/tex]
So, [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] rewritten in the simplest radical form is:
[tex]\[
\sqrt{x}
\][/tex]
