Simplify:
$ \left(4^{-1}+3^{-1}+6^{-2}\right)^{-1} $

Given:

\( \left(4^{-1}+3^{-1}+6^{-2}\right)^{-1} \)

To do:

We have to simplify \( \left(4^{-1}+3^{-1}+6^{-2}\right)^{-1} \).

Solution:

We know that,

$a^{-m}=\frac{1}{a^m}$

$a^m \times a^n=a^{m+n}$

$a^{m}\div a^{n}=a^{m-n}$

Therefore,

$(4^{-1}+3^{-1}+6^{-2})^{-1}=(\frac{1}{4}+\frac{1}{3}+\frac{1}{6^2})^{-1}$

$=(\frac{1}{4}+\frac{1}{3}+\frac{1}{36})^{-1}$

$=(\frac{9+12+1}{36})^{-1}$

$=(\frac{22}{36})^{-1}$

$=(\frac{11}{18})^{-1}$

$=\frac{18}{11}$

Hence, $(4^{-1}+3^{-1}+6^{-2})^{-1}=\frac{18}{11}$.