Answer :
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]
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]
