times-7%5E(2))%2F(5%5E(2)times-7%5E(-4)))%5E(7%2F2)times((5%5E(-2)times-7%5E(3))%2F(5%5E(3)times7%5E(-5)))%5E(-5%2F2).jpeg "((5^(-1)times 7^(2))/(5^(2)times 7^(-4)))^(7/2)times((5^(-2)times 7^(3))/(5^(3)times7^(-5)))^(-5/2)")
Simplifying the Given Expression
The given expression is:
$\left(\frac{5^{-1} \times 7^{2}}{5^{2} \times 7^{-4}}\right)^{7/2} \times \left(\frac{5^{-2} \times 7^{3}}{5^{3} \times 7^{-5}}\right)^{-5/2}$
To simplify this expression, we can start by simplifying each fraction separately.
Simplifying the First Fraction
$\frac{5^{-1} \times 7^{2}}{5^{2} \times 7^{-4}} = \frac{7^{2}}{5^{3} \times 7^{-4}} = \frac{7^{2}}{5^{3}} \times 7^{4} = \frac{1}{5^{3}} \times 7^{6} = \frac{7^{6}}{5^{3}}$
Simplifying the Second Fraction
$\frac{5^{-2} \times 7^{3}}{5^{3} \times 7^{-5}} = \frac{7^{3}}{5^{5} \times 7^{-5}} = \frac{7^{3}}{5^{5}} \times 7^{5} = \frac{7^{8}}{5^{5}}$
Simplifying the Entire Expression
Now, we can simplify the entire expression:
$\left(\frac{7^{6}}{5^{3}}\right)^{7/2} \times \left(\frac{7^{8}}{5^{5}}\right)^{-5/2}$
Using the rule of exponents, we can simplify further:
$\left(\frac{7^{6}}{5^{3}}\right)^{7/2} = \frac{7^{21}}{5^{21/2}}$
$\left(\frac{7^{8}}{5^{5}}\right)^{-5/2} = \frac{5^{25/2}}{7^{20}}$
Final Simplification
Now, we can multiply the two expressions:
$\frac{7^{21}}{5^{21/2}} \times \frac{5^{25/2}}{7^{20}} = \frac{7^{1}}{5^{-2}} = 5^{2} \times 7$
Therefore, the simplified expression is:
$\boxed{25 \times 7 = 175}$
