To rewrite the expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] in its simplest radical form, let's go through the steps to simplify it:
1. Simplify the exponent:
The given exponent is [tex]\(-\frac{3}{6}\)[/tex]. We can simplify this fraction:
[tex]\[
-\frac{3}{6} = -\frac{1}{2}
\][/tex]
2. Rewrite the expression with the simplified exponent:
Substitute the simplified exponent back into the expression:
[tex]\[
\frac{1}{x^{-\frac{1}{2}}}
\][/tex]
3. Utilize the property of exponents:
Recall that [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex], and in reverse, [tex]\(\frac{1}{a^{-b}} = a^b\)[/tex]. Therefore,
[tex]\[
\frac{1}{x^{-\frac{1}{2}}} = x^{\frac{1}{2}}
\][/tex]
4. Expressing in radical form:
Recall the definition of fractional exponents: [tex]\(a^{\frac{1}{n}} = \sqrt[n]{a}\)[/tex]. For our expression, [tex]\(x^{\frac{1}{2}}\)[/tex] can be written as:
[tex]\[
x^{\frac{1}{2}} = \sqrt{x}
\][/tex]
Therefore, the expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] in simplest radical form is [tex]\(\sqrt{x}\)[/tex].
