Question

If $\frac{2^{10}}{5^2}\times \frac{2^2}{5^{10}}+{\left(\frac{5}{2}\right)}^{-12}=\frac{2^a}{5^b}$.Find $a+b$

Answer 1

$\frac{2^{10}}{5^2}\times \frac{2^2}{5^{10}}+{\left(\frac{5}{2}\right)}^{-12}=\frac{2^a}{5^b}$

$\Rightarrow \frac{2^{10}}{5^2}\times \frac{2^2}{5^{10}}+\frac{1}{{\left(\frac{5}{2}\right)}^{12}}=\frac{2^a}{5^b}$ (since $a^{-m}=\frac{1}{a^m}$)

$\Rightarrow \frac{2^{10+2}}{5^{2+10}}+{\left(\frac{2}{5}\right)}^{12}=\frac{2^a}{5^b}$ (Since $a^m\times a^n=a^{m+n}$)

$\Rightarrow \frac{2^{12}}{5^{12}}+\frac{2^{12}}{5^{12}}=\frac{2^a}{5^b}$

$\Rightarrow 2.\frac{2^{12}}{5^{12}}=\frac{2^a}{5^b}$

$\Rightarrow \frac{2^{13}}{5^{12}}=\frac{2^a}{5^b}$

On comparing both sides, we have

$a=13$

$b=12$

$\therefore a+b=13+12=25$

Hence, the correct answer is 25.