Simplify: $ \frac{7^{-5}}{5^{-3}} \times 10^{-4} \times \frac{6^{-5}}{(42)^{-6}} $

Given:

\( \frac{7^{-5}}{5^{-3}} \times 10^{-4} \times \frac{6^{-5}}{(42)^{-6}} \)

To do:

We have to simplify the given expression.

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

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

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

$a^{0}=1$

Therefore,

$ \begin{array}{l}
\frac{7^{-5}}{5^{-3}} \times 10^{-4} \times \frac{6^{-5}}{( 42)^{-6}} =\frac{7^{-5} \times ( 2\times 5)^{-4} \times 6^{-5}}{5^{-3} \times ( 6\times 7)^{-6}}\\
=\frac{7^{-5} \times 2^{-4} \times 5^{-4} \times 6^{-5}}{5^{-3} \times 6^{-6} \times 7^{-6}}\\
=7^{( -5+6)} \times 2^{-4} \times 5^{( -4+3)} \times 6^{( -5+6)}\\
=7^{1} \times 2^{-4} \times 5^{-1} \times 6^{1}\\
=\frac{42}{2^{4} \times 5}\\
=\frac{42}{16\times 5}\\
=\frac{21}{8\times 5}\\
=\frac{21}{40}
\end{array}$