Let's break down the problem step by step:
1. Determine the reciprocals of the given fractions:
- For [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\left(\frac{1}{3}\right)^{-1} = 3
\][/tex]
- For [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^{-1} = 2
\][/tex]
- For [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[
\left(\frac{1}{6}\right)^{-1} = 6
\][/tex]
2. Sum the reciprocals:
[tex]\[
3 + 2 + 6 = 11
\][/tex]
3. Calculate the reciprocal of the sum:
[tex]\[
\left(11\right)^{-1} = \frac{1}{11}
\][/tex]
Therefore, the value of [tex]\(\left\{\left(\frac{1}{3}\right)^{-1}+\left(\frac{1}{2}\right)^{-1}+\left(\frac{1}{6}\right)^{-1}\right\}^{-1}\)[/tex] is:
[tex]\[
\boxed{\frac{1}{11}}
\][/tex]
