Simplify:
(i) $ \left(4^{-1} \times 3^{-1}\right)^{2} $
(ii) $ \left(5^{-1}\div6^{-1}\right)^{3} $
(iii) $ \left(2^{-1}+3^{-1}\right)^{-1} $
(iv) $ \left(3^{-1} \times 4^{-1}\right)^{-1} \times 5^{-1} $

To do:  

We have to simplify the given expressions.

Solution:

We know that,

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

Therefore,

(i) $(4^{-1} \times 3^{-1})^{2}=(\frac{1}{4} \times \frac{1}{3})^{2}$

$=(\frac{1}{12})^{2}$

$=\frac{1}{12} \times \frac{1}{12}$

$=\frac{1}{144}$

(ii) $(5^{-1} \div 6^{-1})^{3}=(\frac{1}{5} \div \frac{1}{6})^{3}$

$=(\frac{1}{5} \times \frac{6}{1})^{3}$

$=(\frac{6}{5})^{3}$

$=\frac{6^3}{5^3}$

$=\frac{216}{125}$

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

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

$=(\frac{5}{6})^{-1}$

$=\frac{6}{5}$

(iv) $(3^{-1} \times 4^{-1})^{-1} \times 5^{-1}=(\frac{1}{3} \times \frac{1}{4})^{-1} \times 5^{-1}$

$=(\frac{1}{12})^{-1} \times \frac{1}{5}$

$=(12)^{1} \times \frac{1}{5}$

$=\frac{12}{5}$