To determine which of the given options is equivalent to [tex]\(36^{-\frac{1}{2}}\)[/tex], we'll proceed step by step.
1. Understanding the expression:
The given expression is [tex]\(36^{-\frac{1}{2}}\)[/tex]. Let's break it down:
- The base is [tex]\(36\)[/tex].
- The exponent is [tex]\(-\frac{1}{2}\)[/tex].
2. Interpreting the exponent:
The exponent [tex]\(-\frac{1}{2}\)[/tex] means two things:
- The negative sign indicates a reciprocal.
- The fraction [tex]\(\frac{1}{2}\)[/tex] indicates a square root.
Therefore, [tex]\(36^{-\frac{1}{2}}\)[/tex] can be rewritten as [tex]\(\frac{1}{36^{\frac{1}{2}}}\)[/tex].
3. Calculating the square root:
We need to find the square root of 36. The square root of 36 is:
[tex]\[
\sqrt{36} = 6
\][/tex]
4. Taking the reciprocal:
Now, taking the reciprocal of 6, we get:
[tex]\[
\frac{1}{6}
\][/tex]
So, [tex]\(36^{-\frac{1}{2}} = \frac{1}{6}\)[/tex].
5. Final Verification:
Comparing this result with the given options:
- [tex]\(-18\)[/tex]
- [tex]\(-6\)[/tex]
- [tex]\(\frac{1}{18}\)[/tex]
- [tex]\(\frac{1}{6}\)[/tex]
We see that [tex]\(\frac{1}{6}\)[/tex] matches our result.
Therefore, the correct answer is [tex]\(\frac{1}{6}\)[/tex].
