​$$\textit{The value of} \left(\frac{15625}{4}\right)^{-\left(\frac{3}{2}\right)} \times \left(\frac{5}{2}\right)^5 \div \sqrt[4]{(625)^{-3}} \quad \text{is:}$$​​

Correct option is D

Given:

​$$\left( \frac{15625}{4} \right)^{-\frac{3}{2}} \times \left( \frac{5}{2} \right)^5 \div \sqrt[4]{(625)^{-3}}$$​​

Solution:

$$\left( \frac{15625}{4} \right)^{-\frac{3}{2}} \times \left( \frac{5}{2} \right)^5 \div \sqrt[4]{(625)^{-3}} \\ \ \\= \left( \frac{(2^2)^{\frac32}}{(125^2)^{\frac{3}{2}}} \right) \times \frac{5^5}{2^5} \div \left( 5^4 \right)^{-\frac{3}{4}} \\ \ \\= \frac{2^3}{125^3} \times \frac{5^5}{2^5} \div \frac{1}{5^3} \\ \ \\= \frac{2^3}{(5^3)^3} \times \frac{5^5}{2^5} \times 5^3 \\ \ \\= \frac{2^3 \times 5^{5+3}}{5^9 \times 2^5} \\ \ \\= \frac{1}{5^{9-8} \times 2^{5-3}} \\ \ \\= \frac{1}{5 \times 2^2} \\ \ \\= \frac{1}{20}$$​