Certainly! Let's solve the given expression step-by-step:
[tex]\[ \frac{1}{\left[\left(3^4\right)^{\frac{1}{2}}\right]^{-2}} \][/tex]
1. Calculate [tex]\(3^4\)[/tex]:
[tex]\[ 3^4 = 3 \times 3 \times 3 \times 3 = 81 \][/tex]
2. Take the square root of [tex]\(81\)[/tex]:
[tex]\[ \left(3^4\right)^{\frac{1}{2}} = 81^{\frac{1}{2}} = \sqrt{81} = 9 \][/tex]
3. Raise the result to the power of [tex]\(-2\)[/tex]:
[tex]\[ \left(9\right)^{-2} = \frac{1}{9^2} = \frac{1}{81} \][/tex]
4. Take the reciprocal of the result:
[tex]\[ \frac{1}{\frac{1}{81}} = 81 \][/tex]
Thus, the value of the given expression is:
[tex]\[ \boxed{81} \][/tex]
