To simplify the given expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex], follow these steps:
1. Simplify the Exponent:
The original expression inside the exponent is [tex]\(-\frac{3}{6}\)[/tex]. Simplify this fraction:
[tex]\[
-\frac{3}{6} = -\frac{1}{2}
\][/tex]
So, the expression can be rewritten as:
[tex]\[
\frac{1}{x^{-\frac{1}{2}}}
\][/tex]
2. Apply the Negative Exponent Rule:
The rule for negative exponents states that [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]. Conversely, [tex]\(\frac{1}{a^{-n}} = a^n\)[/tex]. Using this rule, convert [tex]\(x^{-\frac{1}{2}}\)[/tex] in the denominator to [tex]\(x^{\frac{1}{2}}\)[/tex] in the numerator:
[tex]\[
\frac{1}{x^{-\frac{1}{2}}} = x^{\frac{1}{2}}
\][/tex]
3. Convert to Radical Form:
The expression [tex]\(x^{\frac{1}{2}}\)[/tex] is equivalent to the square root of [tex]\(x\)[/tex]. Using the radical notation, we can rewrite [tex]\(x^{\frac{1}{2}}\)[/tex] as:
[tex]\[
x^{\frac{1}{2}} = \sqrt{x}
\][/tex]
Therefore, the simplest radical form of the given expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] is:
[tex]\[
\sqrt{x}
\][/tex]
