Step 1: Apply the negative exponent rule $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$. $$ \left(\frac{4}{5}\right)^2 \times 5^4 \times \left(\frac{5}{2}\right)^2 \div \left(\frac{2}{5}\right)^3 $$ Step 2: Apply the exponent rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ and express $4$ as $2^2$. $$ \frac{4^2}{5^2} \times 5^4 \times \frac{5^2}{2^2} \div \frac{2^3}{5^3} $$ $$ \frac{(2^2)^2}{5^2} \times 5^4 \times \frac{5^2}{2^2} \div \frac{2^3}{5^3} $$ $$ \frac{2^4}{5^2} \times 5^4 \times \frac{5^2}{2^2} \div \frac{2^3}{5^3} $$ Step 3: Convert the division to multiplication by taking the reciprocal of the last term. $$ \frac{2^4}{5^2} \times 5^4 \times \frac{5^2}{2^2} \times \frac{5^3}{2^3} $$ Step 4: Group terms with the same base and apply the exponent rules $a^m \times a^n = a^{m+n}$ and $\frac{a^m}{a^n} = a^{m-n}$. For base 2: $$ \frac{2^4}{2^2 \times 2^3} = \frac{2^4}{2^{2+3}} = \frac{2^4}{2^5} = 2^{4-5} = 2^{-1} $$ For base 5: $$ \frac{5^4 \times 5^2 \times 5^3}{5^2} = \frac{5^{4+2+3}}{5^2} = \frac{5^9}{5^2} = 5^{9-2} = 5^7 $$ Step 5: Combine the simplified terms. $$ 2^{-1} \times 5^7 = \frac{1}{2} \times 5^7 $$ Step 6: Calculate $5^7$. $$ 5^7 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 78125 $$ Step 7: Substitute the value and simplify. $$ \frac{1}{2} \times 78125 = \frac{78125}{2} $$ The final answer is $\boxed{\frac{78125}{2}}$. Send me the next one 📸