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Simplify:$ \left(\frac{5^{-1} \times 7^{2}}{5^{2} \times 7^{-4}}\right)^{\frac{7}{2}} \times\left(\frac{5^{-2} \times 7^{3}}{5^{3} \times 7^{-5}}\right)^{\frac{-5}{2}} $
Given:
\( \left(\frac{5^{-1} \times 7^{2}}{5^{2} \times 7^{-4}}\right)^{\frac{7}{2}} \times\left(\frac{5^{-2} \times 7^{3}}{5^{3} \times 7^{-5}}\right)^{\frac{-5}{2}} \)
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,
$(\frac{5^{-1} \times 7^{2}}{5^{2} \times 7^{-4}})^{\frac{7}{2}} \times (\frac{5^{-2} \times 7^{3}}{5^{3} \times 7^{-5}})^{\frac{-5}{2}}=\frac{5^{-1 \times \frac{7}{2}} \times 7^{2 \times \frac{7}{2}}}{5^{2 \times \frac{7}{2}} \times 7^{-4 \times \frac{7}{2}}} \times \frac{5^{-2 \times(\frac{-5}{2})}\times 7^{ 3 \times(\frac{-5}{2})}}{5^{3 \times(\frac{-5}{2})} \times 7^{-5 \times(\frac{-5}{2})}}$
$=\frac{5^{\frac{-7}{2}} \times 7^{7}}{5^{7} \times 7^{-14}}$
$=\frac{5^{5} \times 7^{\frac{-15}{2}}}{5^{\frac{-15}{2}} \times 7^{\frac{25}{2}}}$
$=5^{\frac{-7}{2}+5-7+\frac{15}{2}} \times 7^{7-\frac{15}{2}+14-\frac{25}{2}}$
$=5^{-2+\frac{8}{2}} \times 7^{21-\frac{40}{2}}$
$=5^{-2+4} \times 7^{21-20}$
$=5^{2} \times 7^{1}$
$=25 \times 7$
$=175$
Hence, $(\frac{5^{-1} \times 7^{2}}{5^{2} \times 7^{-4}})^{\frac{7}{2}} \times (\frac{5^{-2} \times 7^{3}}{5^{3} \times 7^{-5}})^{\frac{-5}{2}}=175$.
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