To solve the given problem, we need to find the value of [tex]\(\sqrt[4]{(81)^{-2}}\)[/tex].
Let's proceed step-by-step:
1. Evaluate the exponentiation [tex]\((81)^{-2}\)[/tex]:
- When a number is raised to a power of [tex]\(-n\)[/tex], it’s equivalent to the reciprocal of that number raised to the power [tex]\(n\)[/tex]. Thus, we have:
[tex]\[
(81)^{-2} = \frac{1}{81^2}
\][/tex]
2. Calculate [tex]\(81^2\)[/tex]:
- [tex]\(81\)[/tex] is [tex]\(9^2\)[/tex] and [tex]\((9^2)^2 = 9^4\)[/tex]:
[tex]\[
81^2 = 9^4 = 6561
\][/tex]
3. Determine the reciprocal:
- Therefore:
[tex]\[
(81)^{-2} = \frac{1}{81^2} = \frac{1}{6561}
\][/tex]
4. Compute the fourth root:
- We need to find the fourth root of [tex]\(\frac{1}{6561}\)[/tex]:
[tex]\[
\sqrt[4]{\frac{1}{6561}}
\][/tex]
- A simpler way to approach this is to recognize:
[tex]\[
6561 = 9^4
\][/tex]
[tex]\[
\sqrt[4]{\frac{1}{9^4}} = \frac{1}{\sqrt[4]{9^4}}
\][/tex]
Since [tex]\(\sqrt[4]{9^4} = 9\)[/tex], we get:
[tex]\[
\sqrt[4]{\frac{1}{6561}} = \frac{1}{9}
\][/tex]
5. Conclusion:
- Therefore, the value is [tex]\(\frac{1}{9}\)[/tex].
So, the correct answer is:
a. [tex]\(\frac{1}{9}\)[/tex]
