Question

Evaluate: ${\left(\frac{4}{5}\right)}^2\times 5^4\times {\left(\frac{2}{5}\right)}^{-2}÷{\left(\frac{5}{2}\right)}^{-3}$

Answer 1

${\left(\frac{4}{5}\right)}^2\times 5^4\times {\left(\frac{2}{5}\right)}^{-2}÷{\left(\frac{5}{2}\right)}^{-3}$

= $\frac{4^2}{5^2}\times 5^4\times \frac{5^2}{2^2}÷\frac{2^3}{5^3}$ [Using $(\frac{a}{b}{)}^{-n}=(\frac{b}{a}{)}^n$

= $\frac{4^2}{5^2}\times 5^4\times \frac{5^2}{2^2}\times \frac{5^3}{2^3}$

= $\frac{4^2}{5^2}\times 5^4\times \frac{5^5}{2^5}$ [Using $a^m\times a^n=a^{m+n}$]

= $\frac{2^4}{5^2}\times \frac{5^9}{2^5}$ [Using $a^m\times a^n=a^{m+n}$ and $4^2=(2^2{)}^2=2^4$]

= $5^{9-2}\times 2^{4-5}$ [Using $a^m÷a^n=a^{m-n}$]

= $\frac{5^7}{2}$

Hence, the required result is $\frac{5^7}{2}$